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Unformatted text preview: 5E-11(pp 686-695) 1/18/06 9:31 AM Page 686 CHAPTER 11 Parametric curves are used to represent letters and other symbols on laser printers. See the Laboratory Project on page 705. Parametric Equations and Polar Coordinates 5E-11(pp 686-695) 1/18/06 9:31 AM Page 687 So far we have described plane curves by giving y as a function of x ͓y ෇ f ͑x͔͒ or x as a function of y ͓x ෇ t͑ y͔͒ or by giving a relation between x and y that defines y implicitly as a function of x ͓ f ͑x, y͒ ෇ 0͔. In this chapter we discuss two new methods for describing curves. Some curves, such as the cycloid, are best handled when both x and y are given in terms of a third variable t called a parameter ͓x ෇ f ͑t͒, y ෇ t͑t͔͒. Other curves, such as the cardioid, have their most convenient description when we use a new coordinate system, called the polar coordinate system. |||| 11.1 y Curves Defined by Parametric Equations C (x, y)={ f(t), g(t)} 0 FIGURE 1 x Imagine that a particle moves along the curve C shown in Figure 1. It is impossible to describe C by an equation of the form y ෇ f ͑x͒ because C fails the Vertical Line Test. But the x- and y-coordinates of the particle are functions of time and so we can write x ෇ f ͑t͒ and y ෇ t͑t͒. Such a pair of equations is often a convenient way of describing a curve and gives rise to the following definition. Suppose that x and y are both given as functions of a third variable t (called a parameter) by the equations x ෇ f ͑t͒ y ෇ t͑t͒ (called parametric equations). Each value of t determines a point ͑x, y͒, which we can plot in a coordinate plane. As t varies, the point ͑x, y͒ ෇ ͑ f ͑t͒, t͑t͒͒ varies and traces out a curve C, which we call a parametric curve. The parameter t does not necessarily represent time and, in fact, we could use a letter other than t for the parameter. But in many applications of parametric curves, t does denote time and therefore we can interpret ͑x, y͒ ෇ ͑ f ͑t͒, t͑t͒͒ as the position of a particle at time t. EXAMPLE 1 Sketch and identify the curve defined by the parametric equations x ෇ t 2 Ϫ 2t y෇tϩ1 SOLUTION Each value of t gives a point on the curve, as shown in the table. For instance, if t ෇ 0, then x ෇ 0, y ෇ 1 and so the corresponding point is ͑0, 1͒. In Figure 2 we plot the points ͑x, y͒ determined by several values of the parameter and we join them to produce a curve. t Ϫ2 Ϫ1 0 1 2 3 4 x 8 3 0 Ϫ1 0 3 8 y y Ϫ1 0 1 2 3 4 5 t=4 t=3 t=2 t=1 (0, 1) 8 t=0 0 x t=_1 t=_2 FIGURE 2 A particle whose position is given by the parametric equations moves along the curve in the direction of the arrows as t increases. Notice that the consecutive points marked on 687 5E-11(pp 686-695) 688 ❙❙❙❙ 1/18/06 9:32 AM Page 688 CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES the curve appear at equal time intervals but not at equal distances. That is because the particle slows down and then speeds up as t increases. It appears from Figure 2 that the curve traced out by the particle may be a parabola. This can be confirmed by eliminating the parameter t as follows. We obtain t ෇ y Ϫ 1 from the second equation and substitute into the first equation. This gives x ෇ t 2 Ϫ 2t ෇ ͑y Ϫ 1͒2 Ϫ 2͑y Ϫ 1͒ ෇ y 2 Ϫ 4y ϩ 3 and so the curve represented by the given parametric equations is the parabola x ෇ y 2 Ϫ 4y ϩ 3. No restriction was placed on the parameter t in Example 1, so we assumed that t could be any real number. But sometimes we restrict t to lie in a finite interval. For instance, the parametric curve y x ෇ t 2 Ϫ 2t (8, 5) 0ഛtഛ4 shown in Figure 3 is the part of the parabola in Example 1 that starts at the point ͑0, 1͒ and ends at the point ͑8, 5͒. The arrowhead indicates the direction in which the curve is traced as t increases from 0 to 4. In general, the curve with parametric equations (0, 1) 0 y෇tϩ1 x x ෇ f ͑t͒ y ෇ t͑t͒ aഛtഛb has initial point ͑ f ͑a͒, t͑a͒͒ and terminal point ͑ f ͑b, t͑b͒͒. FIGURE 3 EXAMPLE 2 What curve is represented by the parametric equations x ෇ cos t, y ෇ sin t, 0 ഛ t ഛ 2␲ ? SOLUTION If we plot points, it appears that the curve is a circle. We can confirm this impression by eliminating t. Observe that x 2 ϩ y 2 ෇ cos 2t ϩ sin 2t ෇ 1 Thus, the point ͑x, y͒ moves on the unit circle x 2 ϩ y 2 ෇ 1. Notice that in this example the parameter t can be interpreted as the angle (in radians) shown in Figure 4. As t increases from 0 to 2␲, the point ͑x, y͒ ෇ ͑cos t, sin t͒ moves once around the circle in the counterclockwise direction starting from the point ͑1, 0͒. π t= 2 y (cos t, sin t ) t=0 t=π t 0 (1, 0) x t=2π t= FIGURE 4 3π 2 EXAMPLE 3 What curve is represented by the parametric equations x ෇ sin 2t, y ෇ cos 2t, 0 ഛ t ഛ 2␲ ? 5E-11(pp 686-695) 1/18/06 9:32 AM Page 689 SECTION 11.1 CURVES DEFINED BY PARAMETRIC EQUATIONS ❙❙❙❙ 689 SOLUTION Again we have x 2 ϩ y 2 ෇ sin 2 2t ϩ cos 2 2t ෇ 1 so the parametric equations again represent the unit circle x 2 ϩ y 2 ෇ 1. But as t increases from 0 to 2␲, the point ͑x, y͒ ෇ ͑sin 2t, cos 2t͒ starts at ͑0, 1͒ and moves twice around the circle in the clockwise direction as indicated in Figure 5. y t=0, π, 2π (0, 1) 0 x FIGURE 5 Examples 2 and 3 show that different sets of parametric equations can represent the same curve. Thus, we distinguish between a curve, which is a set of points, and a parametric curve, in which the points are traced in a particular way. (_1, 1) y (1, 1) EXAMPLE 4 Sketch the curve with parametric equations x ෇ sin t, y ෇ sin 2t. SOLUTION Observe that y ෇ ͑sin t͒ 2 ෇ x 2 and so the point ͑x, y͒ moves on the parabola 0 x FIGURE 6 y ෇ x 2. But note also that, since Ϫ1 ഛ sin t ഛ 1, we have Ϫ1 ഛ x ഛ 1, so the parametric equations represent only the part of the parabola for which Ϫ1 ഛ x ഛ 1. Since sin t is periodic, the point ͑x, y͒ ෇ ͑sin t, sin 2t͒ moves back and forth infinitely often along the parabola from ͑Ϫ1, 1͒ to ͑1, 1͒. (See Figure 6.) x x=cos t t Module 11.1A gives an animation of the relationship between motion along a parametric curve x ෇ f ͑t͒, y ෇ t͑t͒ and motion along the graphs of f and t as functions of t. Clicking on TRIG gives you the family of parametric curves x ෇ a cos bt y ෇ c sin dt If you choose a ෇ b ෇ c ෇ d ෇ 1 and click START, you will see how the graphs of x ෇ cos t and y ෇ sin t relate to the circle in Example 2. If you choose a ෇ b ෇ c ෇ 1, d ෇ 2, you will see graphs as in Figure 7. By clicking on PAUSE and then repeatedly on STEP, you can see from the color coding how motion along the graphs of x ෇ cos t and y ෇ sin 2t corresponds to motion along the parametric curve, which is called a Lissajous figure. y y x FIGURE 7 x=cos t y=sin 2t t y=sin 2t 5E-11(pp 686-695) 690 ❙❙❙❙ 1/18/06 9:32 AM Page 690 CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES Graphing Devices Most graphing calculators and computer graphing programs can be used to graph curves defined by parametric equations. In fact, it’s instructive to watch a parametric curve being drawn by a graphing calculator because the points are plotted in order as the corresponding parameter values increase. 3 EXAMPLE 5 Use a graphing device to graph the curve x ෇ y 4 Ϫ 3y 2. SOLUTION If we let the parameter be t ෇ y, then we have the equations _3 x ෇ t 4 Ϫ 3t 2 3 _3 y෇t Using these parametric equations to graph the curve, we obtain Figure 8. It would be possible to solve the given equation ͑x ෇ y 4 Ϫ 3y 2 ͒ for y as four functions of x and graph them individually, but the parametric equations provide a much easier method. FIGURE 8 In general, if we need to graph an equation of the form x ෇ t͑y͒, we can use the parametric equations x ෇ t͑t͒ y෇t Notice also that curves with equations y ෇ f ͑x͒ (the ones we are most familiar with— graphs of functions) can also be regarded as curves with parametric equations x෇t y ෇ f ͑t͒ Graphing devices are particularly useful when sketching complicated curves. For instance, the curves shown in Figures 9 and 10 would be virtually impossible to produce by hand. 8 _6.5 3 6.5 _1.5 _8 1.5 _3 FIGURE 9 FIGURE 10 x=t+2 sin 2t, y=t+2 cos 5t, _2π р t р 2π x=cos t-cos 80t sin t, y=2 sin t-sin 80t, 0 р t р 2π One of the most important uses of parametric curves is in computer-aided design (CAD). In the Laboratory Project after Section 10.2 we will investigate special parametric curves, called Bézier curves, that are used extensively in manufacturing, especially in the automotive industry. These curves are also employed in specifying the shapes of letters and other symbols in laser printers. The Cycloid An animation in Module 11.1B shows how the cycloid is formed as the circle moves. EXAMPLE 6 The curve traced out by a point P on the circumference of a circle as the circle rolls along a straight line is called a cycloid (see Figure 11). If the circle has radius r and rolls along the x-axis and if one position of P is the origin, find parametric equations for the cycloid. 5E-11(pp 686-695) 1/18/06 9:32 AM Page 691 SECTION 11.1 CURVES DEFINED BY PARAMETRIC EQUATIONS ❙❙❙❙ 691 P P FIGURE 11 SOLUTION We choose as parameter the angle of rotation ␪ of the circle ͑␪ ෇ 0 when P is at the origin). Suppose the circle has rotated through ␪ radians. Because the circle has been in contact with the line, we see from Figure 12 that the distance it has rolled from the origin is OT ෇ arc PT ෇ r␪ y r P Խ Խ C (r¨, r) ¨ P Therefore, the center of the circle is C͑r␪, r͒. Let the coordinates of P be ͑x, y͒. Then from Figure 12 we see that Q y x T O Խ Խ Խ Խ y ෇ Խ TC Խ Ϫ Խ QC Խ ෇ r Ϫ r cos ␪ ෇ r͑1 Ϫ cos ␪ ͒ x ෇ OT Ϫ PQ ෇ r ␪ Ϫ r sin ␪ ෇ r͑␪ Ϫ sin ␪ ͒ x r¨ FIGURE 12 Therefore, parametric equations of the cycloid are 1 x ෇ r͑␪ Ϫ sin ␪ ͒ y ෇ r͑1 Ϫ cos ␪ ͒ ␪ʦ‫ޒ‬ One arch of the cycloid comes from one rotation of the circle and so is described by 0 ഛ ␪ ഛ 2␲. Although Equations 1 were derived from Figure 12, which illustrates the case where 0 Ͻ ␪ Ͻ ␲͞2, it can be seen that these equations are still valid for other values of ␪ (see Exercise 37). Although it is possible to eliminate the parameter ␪ from Equations 1, the resulting Cartesian equation in x and y is very complicated and not as convenient to work with as the parametric equations. A cycloid B FIGURE 13 P P P P P FIGURE 14 One of the first people to study the cycloid was Galileo, who proposed that bridges be built in the shape of cycloids and who tried to find the area under one arch of a cycloid. Later this curve arose in connection with the brachistochrone problem: Find the curve along which a particle will slide in the shortest time (under the influence of gravity) from a point A to a lower point B not directly beneath A. The Swiss mathematician John Bernoulli, who posed this problem in 1696, showed that among all possible curves that join A to B, as in Figure 13, the particle will take the least time sliding from A to B if the curve is part of an inverted arch of a cycloid. The Dutch physicist Huygens had already shown that the cycloid is also the solution to the tautochrone problem; that is, no matter where a particle P is placed on an inverted cycloid, it takes the same time to slide to the bottom (see Figure 14). Huygens proposed that pendulum clocks (which he invented) swing in cycloidal arcs because then the pendulum takes the same time to make a complete oscillation whether it swings through a wide or a small arc. Families of Parametric Curves EXAMPLE 7 Investigate the family of curves with parametric equations x ෇ a ϩ cos t y ෇ a tan t ϩ sin t What do these curves have in common? How does the shape change as a increases? 5E-11(pp 686-695) 692 ❙❙❙❙ 1/18/06 9:32 AM Page 692 CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES SOLUTION We use a graphing device to produce the graphs for the cases a ෇ Ϫ2, Ϫ1, Ϫ0.5, Ϫ0.2, 0, 0.5, 1, and 2 shown in Figure 15. Notice that all of these curves (except the case a ෇ 0) have two branches, and both branches approach the vertical asymptote x ෇ a as x approaches a from the left or right. a=_2 a=_1 a=0 a=0.5 x=a+cos t, y=a tan t+sin t, all graphed in the viewing rectangle ͓_4, 4͔ by ͓_4, 4͔ |||| 11.1 a=1 8. x ෇ 1 ϩ 3t, |||| 1. x ෇ 1 ϩ st, y ෇ t 2 Ϫ 4 t, 2. x ෇ 2 cos t, y ෇ t Ϫ cos t, 3. x ෇ 5 sin t, y ෇ t 2, Ϫt 4. x ෇ e ϩ t, ■ 5–10 a=2 Exercises Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases. ■ a=_0.2 When a Ͻ Ϫ1, both branches are smooth; but when a reaches Ϫ1, the right branch acquires a sharp point, called a cusp. For a between Ϫ1 and 0 the cusp turns into a loop, which becomes larger as a approaches 0. When a ෇ 0, both branches come together and form a circle (see Example 2). For a between 0 and 1, the left branch has a loop, which shrinks to become a cusp when a ෇ 1. For a Ͼ 1, the branches become smooth again, and as a increases further, they become less curved. Notice that the curves with a positive are reflections about the y-axis of the corresponding curves with a negative. These curves are called conchoids of Nicomedes after the ancient Greek scholar Nicomedes. He called them conchoids because the shape of their outer branches resembles that of a conch shell or mussel shell. FIGURE 15 Members of the family 1–4 a=_0.5 ■ ■ ■ 10. x ෇ t , 0ഛtഛ5 0 ഛ t ഛ 2␲ ■ 11–18 Ϫ2 ഛ t ഛ 2 ■ ■ ■ ■ ■ |||| (a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases. (b) Eliminate the parameter to find a Cartesian equation of the curve. 5. x ෇ 3t Ϫ 5, 6. x ෇ 1 ϩ t, 7. x ෇ t 2 Ϫ 2, y ෇ 2t ϩ 1 y ෇ 5 Ϫ 2t, Ϫ2 ഛ t ഛ 3 y ෇ 5 Ϫ 2t, Ϫ3 ഛ t ഛ 4 y෇1Ϫt y ෇ t3 2 Ϫ␲ ഛ t ഛ ␲ y ෇ e t Ϫ t, ■ 9. x ෇ st, ■ ■ y ෇ 2 Ϫ t2 ■ ■ ■ ■ ■ ■ ■ ■ ■ |||| (a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. 11. x ෇ sin ␪, y ෇ cos ␪, 0 ഛ ␪ ഛ ␲ 12. x ෇ 4 cos ␪, y ෇ 5 sin ␪, Ϫ␲͞2 ഛ ␪ ഛ ␲͞2 13. x ෇ sin ␪, y ෇ cos ␪ 14. x ෇ sec ␪, y ෇ tan ␪, Ϫ␲͞2 Ͻ ␪ Ͻ ␲͞2 2 15. x ෇ e t, 16. x ෇ ln t, 2 y ෇ e Ϫt y ෇ st, tജ1 ■ 5E-11(pp 686-695) 1/18/06 9:33 AM Page 693 ❙❙❙❙ SECTION 11.1 CURVES DEFINED BY PARAMETRIC EQUATIONS 17. x ෇ cosh t, y ෇ sinh t 18. x ෇ 1 ϩ cos ␪, ■ ■ ■ 25–27 |||| Use the graphs of x ෇ f ͑t͒ and y ෇ t͑t͒ to sketch the parametric curve x ෇ f ͑t͒, y ෇ t͑t͒. Indicate with arrows the direction in which the curve is traced as t increases. y ෇ 2 cos ␪ Ϫ 1 ■ ■ ■ ■ ■ ■ ■ ■ ■ 25. x y Describe the motion of a particle with position ͑x, y͒ as t varies in the given interval. 19–22 |||| 19. x ෇ cos ␲ t, y ෇ sin ␲ t, 20. x ෇ 2 ϩ cos t, 21. x ෇ 2 sin t, ■ 1 ■ y ෇ 3 cos t , 0 ഛ t ഛ 2␲ ■ ■ ■ ■ t 1 1 x ■ t y 1 26. ■ t _1 0 ഛ t ഛ 2␲ 1 y ෇ cos t, 0 ഛ t ഛ 4␲ 22. x ෇ cos 2t, ■ 1 1ഛtഛ2 y ෇ 3 ϩ sin t, 693 ■ ■ ■ 1 t 23. Suppose a curve is given by the parametric equations x ෇ f ͑t͒, y ෇ t͑t͒, where the range of f is ͓1, 4͔ and the range of t is ͓2 , 3͔. What can you say about the curve? 27. x y 1 1 24. Match the graphs of the parametric equations x ෇ f ͑t͒ and y ෇ t͑t͒ in (a)–(d) with the parametric curves labeled I–IV. Give reasons for your choices. (a) 1 1 t t I y x 2 y 2 1 ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 28. Match the parametric equations with the graphs labeled I–VI. 1 1 1 t 2 x t (b) II x y y 2 2 2 Give reasons for your choices. (Do not use a graphing device.) (a) x ෇ t 3 Ϫ 2t, y ෇ t 2 Ϫ t (b) x ෇ t 3 Ϫ 1, y ෇ 2 Ϫ t 2 (c) x ෇ sin 3t, y ෇ sin 4t (d) x ෇ t ϩ sin 2t, y ෇ t ϩ sin 3t (e) x ෇ sin͑t ϩ sin t͒, y ෇ cos͑t ϩ cos t͒ (f) x ෇ cos t, y ෇ sin͑t ϩ sin 5t͒ I 1t 1t y 2 x x 0 (c) y II x 0 III x y y 2 2 2 t 1 III 1 2 t IV y y 2 x 0 x x 0 (d) x IV y y 0 2 t VI y 2 2 2 V y x 2 t 0 2 x 3 5 ; 29. Graph the curve x ෇ y Ϫ 3y ϩ y . x 5E-11(pp 686-695) 694 ❙❙❙❙ 1/18/06 9:33 AM Page 694 CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES 5 2 ; 30. Graph the curves y ෇ x and x ෇ y͑ y Ϫ 1͒ and find their points of intersection correct to one decimal place. 31. (a) Show that the parametric equations x ෇ x 1 ϩ ͑x 2 Ϫ x 1 ͒t figure, using the angle ␪ as the parameter. Then eliminate the parameter and identify the curve. 40. If a and b are fixed numbers, find parametric equations for the y ෇ y1 ϩ ͑ y 2 Ϫ y1 ͒t where 0 ഛ t ഛ 1, describe the line segment that joins the points P1͑x 1, y1 ͒ and P2͑x 2 , y 2 ͒. (b) Find parametric equations to represent the line segment from ͑Ϫ2, 7͒ to ͑3, Ϫ1͒. curve that consists of all possible positions of the point P in the figure, using the angle ␪ as the parameter. The line segment AB is tangent to the larger circle. y A ; 32. Use a graphing device and the result of Exercise 31(a) to draw the triangle with vertices A ͑1, 1͒, B ͑4, 2͒, and C ͑1, 5͒. a P b ¨ 33. Find parametric equations for the path of a particle that moves O along the circle x 2 ϩ ͑ y Ϫ 1͒2 ෇ 4 in the manner described. (a) Once around clockwise, starting at ͑2, 1͒ (b) Three times around counterclockwise, starting at ͑2, 1͒ (c) Halfway around counterclockwise, starting at ͑0, 3͒ x B ; 34. Graph the semicircle traced by the particle in Exercise 33(c). ; 35. (a) Find parametric equations for the ellipse x 2͞a 2 ϩ y 2͞b 2 ෇ 1. [Hint: Modify the equations of a circle in Example 2.] (b) Use these parametric equations to graph the ellipse when a ෇ 3 and b ෇ 1, 2, 4, and 8. (c) How does the shape of the ellipse change as b varies? 36. Find three different sets of parametric equations to represent the curve y ෇ x , x ʦ ‫.ޒ‬ 3 41. A curve, called a witch of Maria Agnesi, consists of all pos- sible positions of the point P in the figure. Show that parametric equations for this curve can be written as x ෇ 2a cot ␪ y ෇ 2a sin 2␪ Sketch the curve. y C y=2a 37. Derive Equations 1 for the case ␲͞2 Ͻ ␪ Ͻ ␲. 38. Let P be a point at a distance d from the center of a circle of A radius r. The curve traced out by P as the circle rolls along a straight line is called a trochoid. (Think of the motion of a point on a spoke of a bicycle wheel.) The cycloid is the special case of a trochoid with d ෇ r. Using the same parameter ␪ as for the cycloid and assuming the line is the x-axis and ␪ ෇ 0 when P is at one of its lowest points, show that parametric equations of the trochoid are x ෇ r␪ Ϫ d sin ␪ y ෇ r Ϫ d cos ␪ Sketch the trochoid for the cases d Ͻ r and d Ͼ r. 39. If a and b are fixed numbers, find parametric equations for the curve that consists of all possible positions of the point P in the y ¨ O x 42. Find parametric equations for the curve that consists of all pos- Խ y ¨ O B x=2a b P P x Խ Խ Խ sible positions of the point P in the figure, where OP ෇ AB . Sketch the curve. (This curve is called the cissoid of Diocles after the Greek scholar Diocles, who introduced the cissoid as a graphical method for constructing the edge of a cube whose volume is twice that of a given cube.) A a P a O a x 5E-11(pp 686-695) 1/18/06 9:33 AM Page 695 LABORATORY PROJECT RUNNING CIRCLES AROUND CIRCLES ; 43. Suppose that the position of one particle at time t is given by x 1 ෇ 3 sin t 0 ഛ t ഛ 2␲ y1 ෇ 2 cos t and the position of a second particle is given by x 2 ෇ Ϫ3 ϩ cos t 0 ഛ t ഛ 2␲ y 2 ෇ 1 ϩ sin t (a) Graph the paths of both particles. How many points of intersection are there? (b) Are any of these points of intersection collision points? In other words, are the particles ever at the same place at the same time? If so, find the collision points. (c) Describe what happens if the path of the second particle is given by x 2 ෇ 3 ϩ cos t 0 ഛ t ഛ 2␲ y 2 ෇ 1 ϩ sin t 44. If a projectile is fired with an initial velocity of v 0 meters per second at an angle ␣ above the horizontal and air resistance is assumed to be negligible, then its position after t seconds is given by the parametric equations x ෇ ͑v 0 cos ␣͒t y ෇ ͑v 0 sin ␣͒t Ϫ 1 tt 2 2 ; 695 (b) Use a graphing device to check your answers to part (a). Then graph the path of the projectile for several other values of the angle ␣ to see where it hits the ground. Summarize your findings. (c) Show that the path is parabolic by eliminating the parameter. ; 45. Investigate the family of curves defined by the parametric equa- tions x ෇ t 2, y ෇ t 3 Ϫ ct. How does the shape change as c increases? Illustrate by graphing several members of the family. ; 46. The swallowtail catastrophe curves are defined by the parametric equations x ෇ 2ct Ϫ 4t 3, y ෇ Ϫct 2 ϩ 3t 4. Graph several of these curves. What features do the curves have in common? How do they change when c increases? ; 47. The curves with equations x ෇ a sin nt, y ෇ b cos t are called Lissajous figures. Investigate how these curves vary when a, b, and n vary. (Take n to be a positive integer.) ; 48. Investigate the family of curves defined by the parametric equations 2 where t is the acceleration due to gravity (9.8 m͞s ). (a) If a gun is fired with ␣ ෇ 30Њ and v 0 ෇ 500 m͞s, when will the bullet hit the ground? How far from the gun will it hit the ground? What is the maximum height reached by the bullet? ❙❙❙❙ x ෇ sin t ͑c Ϫ sin t͒ y ෇ cos t ͑c Ϫ sin t͒ How does the shape change as c changes? In particular, you should identify the transitional values of c for which the basic shape of the curve changes. LABORATORY PROJECT ; Running Circles around Circles In this project we investigate families of curves, called hypocycloids and epicycloids, that are generated by the motion of a point on a circle that rolls inside or outside another circle. y 1. A hypocycloid is a curve traced out by a fixed point P on a circle C of radius b as C rolls on the inside of a circle with center O and radius a. Show that if the initial position of P is ͑a, 0͒ and the parameter ␪ is chosen as in the figure, then parametric equations of the hypocycloid are C b ¨ a O P (a, 0) A x ͩ x ෇ ͑a Ϫ b͒ cos ␪ ϩ b cos aϪb ␪ b ͪ ͩ y ෇ ͑a Ϫ b͒ sin ␪ Ϫ b sin aϪb ␪ b ͪ 2. Use a graphing device (or the interactive graphic in TEC Module 11.1B) to draw the graphs of hypocycloids with a a positive integer and b ෇ 1. How does the value of a affect the graph? Show that if we take a ෇ 4, then the parametric equations of the hypocycloid reduce to x ෇ 4 cos 3␪ Look at Module 11.1B to see how hypocycloids and epicycloids are formed by the motion of rolling circles. y ෇ 4 sin 3␪ This curve is called a hypocycloid of four cusps, or an astroid. 3. Now try b ෇ 1 and a ෇ n͞d, a fraction where n and d have no common factor. First let n ෇ 1 and try to determine graphically the effect of the denominator d on the shape of the graph. Then let n vary while keeping d constant. What happens when n ෇ d ϩ 1? 5E-11(pp 696-705) 696 ❙❙❙❙ 1/18/06 9:54 AM Page 696 CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES 4. What happens if b ෇ 1 and a is irrational? Experiment with an irrational number like s2 or e Ϫ 2. Take larger and larger values for ␪ and speculate on what would happen if we were to graph the hypocycloid for all real values of ␪. 5. If the circle C rolls on the outside of the fixed circle, the curve traced out by P is called an epicycloid. Find parametric equations for the epicycloid. 6. Investigate the possible shapes for epicycloids. Use methods similar to Problems 2–4. |||| 11.2 Calculus with Parametric Curves Having seen how to represent curves by parametric equations, we now apply the methods of calculus to these parametric curves. In particular, we solve problems involving tangents, area, arc length, and surface area. Tangents In the preceding section we saw that some curves defined by parametric equations x ෇ f ͑t͒ and y ෇ t͑t͒ can also be expressed, by eliminating the parameter, in the form y ෇ F͑x͒. (See Exercise 67 for general conditions under which this is possible.) If we substitute x ෇ f ͑t͒ and y ෇ t͑t͒ in the equation y ෇ F͑x͒, we get t͑t͒ ෇ F͑ f ͑t͒͒ and so, if t, F , and f are differentiable, the Chain Rule gives tЈ͑t͒ ෇ FЈ͑ f ͑t͒͒f Ј͑t͒ ෇ FЈ͑x͒f Ј͑t͒ If f Ј͑t͒ 1 0, we can solve for FЈ͑x͒: FЈ͑x͒ ෇ tЈ͑t͒ f Ј͑t͒ Since the slope of the tangent to the curve y ෇ F͑x͒ at ͑x, F͑x͒͒ is FЈ͑x͒, Equation 1 enables us to find tangents to parametric curves without having to eliminate the parameter. Using Leibniz notation, we can rewrite Equation 1 in an easily remembered form: |||| If we think of a parametric curve as being traced out by a moving particle, then dy͞dt and dx͞dt are the vertical and horizontal velocities of the particle and Formula 2 says that the slope of the tangent is the ratio of these velocities. 2 dy dy dt ෇ dx dx dt if dx dt 0 It can be seen from Equation 2 that the curve has a horizontal tangent when dy͞dt ෇ 0 (provided that dx͞dt 0 ) and it has a vertical tangent when dx͞dt ෇ 0 (provided that dy͞dt 0). This information is useful for sketching parametric curves. 5E-11(pp 696-705) 1/18/06 9:55 AM Page 697 SECTION 11.2 CALCULUS WITH PARAMETRIC CURVES ❙❙❙❙ 697 As we know from Chapter 4, it is also useful to consider d 2 y͞dx 2. This can be found by replacing y by dy͞dx in Equation 2: | 2 d y Note that dx 2 d 2y dt 2 d 2x dt 2 d2y d 2 ෇ dx dx ͩ ͪ ͩ ͪ dy dx d dt ෇ dy dx dx dt EXAMPLE 1 A curve C is defined by the parametric equations x ෇ t 2, y ෇ t 3 Ϫ 3t. (a) Show that C has two tangents at the point (3, 0) and find their equations. (b) Find the points on C where the tangent is horizontal or vertical. (c) Determine where the curve is concave upward or downward. (d) Sketch the curve. SOLUTION (a) Notice that y ෇ t 3 Ϫ 3t ෇ t͑t 2 Ϫ 3͒ ෇ 0 when t ෇ 0 or t ෇ Ϯs3. Therefore, the point ͑3, 0͒ on C arises from two values of the parameter, t ෇ s3 and t ෇ Ϫs3. This indicates that C crosses itself at ͑3, 0͒. Since dy dy͞dt 3t 2 Ϫ 3 3 ෇ ෇ ෇ dx dx͞dt 2t 2 ͩ ͪ tϪ 1 t the slope of the tangent when t ෇ Ϯs3 is dy͞dx ෇ Ϯ6͞(2s3 ) ෇ Ϯs3 , so the equations of the tangents at ͑3, 0͒ are y ෇ s3 ͑x Ϫ 3͒ y y=œ„ (x-3) 3 t=_1 (1, 2) (3, 0) 0 (b) C has a horizontal tangent when dy͞dx ෇ 0, that is, when dy͞dt ෇ 0 and dx͞dt 0. Since dy͞dt ෇ 3t 2 Ϫ 3, this happens when t 2 ෇ 1, that is, t ෇ Ϯ1. The corresponding points on C are ͑1, Ϫ2͒ and (1, 2). C has a vertical tangent when dx͞dt ෇ 2t ෇ 0, that is, t ෇ 0. (Note that dy͞dt 0 there.) The corresponding point on C is (0, 0). (c) To determine concavity we calculate the second derivative: x 2 d y ෇ dx 2 t=1 (1, _2) y=_ œ„ (x-3) 3 FIGURE 1 y ෇ Ϫs3 ͑x Ϫ 3͒ and d dt ͩ ͪ ͩ ͪ dy dx dx dt ෇ 3 2 1ϩ 2t 1 t2 ෇ 3͑t 2 ϩ 1͒ 4t 3 Thus, the curve is concave upward when t Ͼ 0 and concave downward when t Ͻ 0. (d) Using the information from parts (b) and (c), we sketch C in Figure 1. EXAMPLE 2 (a) Find the tangent to the cycloid x ෇ r͑␪ Ϫ sin ␪ ͒, y ෇ r͑1 Ϫ cos ␪ ͒ at the point where ␪ ෇ ␲͞3. (See Example 6 in Section 11.1.) (b) At what points is the tangent horizontal? When is it vertical? SOLUTION (a) The slope of the tangent line is dy dy͞d␪ r sin ␪ sin ␪ ෇ ෇ ෇ dx dx͞d␪ r͑1 Ϫ cos ␪ ͒ 1 Ϫ cos ␪ 5E-11(pp 696-705) 698 ❙❙❙❙ 1/18/06 9:55 AM Page 698 CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES When ␪ ෇ ␲͞3, we have ͩ x෇r ␲ ␲ Ϫ sin 3 3 ͪ ͩ ෇r ␲ s3 Ϫ 3 2 ͪ ͩ y ෇ r 1 Ϫ cos ␲ 3 ͪ ෇ r 2 dy sin͑␲͞3͒ s3͞2 ෇ ෇ ෇ s3 dx 1 Ϫ cos͑␲͞3͒ 1Ϫ1 2 and Therefore, the slope of the tangent is s3 and its equation is yϪ ͩ r r␲ rs3 ෇ s3 x Ϫ ϩ 2 3 2 ͪ or ͩ s3 x Ϫ y ෇ r ͪ ␲ Ϫ2 s3 The tangent is sketched in Figure 2. (_πr, 2r) y (πr, 2r) (3πr, 2r) (5πr, 2r) π ¨= 3 0 FIGURE 2 2πr 4πr x (b) The tangent is horizontal when dy͞dx ෇ 0, which occurs when sin ␪ ෇ 0 and 1 Ϫ cos ␪ 0, that is, ␪ ෇ ͑2n Ϫ 1͒␲, n an integer. The corresponding point on the cycloid is ͑͑2n Ϫ 1͒␲ r, 2r͒. When ␪ ෇ 2n␲, both dx͞d␪ and dy͞d␪ are 0. It appears from the graph that there are vertical tangents at those points. We can verify this by using l’Hospital’s Rule as follows: lim ␪ l2n␲ ϩ dy sin ␪ cos ␪ ෇ lim ϩ ෇ lim ϩ ෇ϱ dx ␪ l2n␲ 1 Ϫ cos ␪ ␪ l2n␲ sin ␪ A similar computation shows that dy͞dx l Ϫϱ as ␪ l 2n␲ Ϫ, so indeed there are vertical tangents when ␪ ෇ 2n␲, that is, when x ෇ 2n␲ r. Areas We know that the area under a curve y ෇ F͑x͒ from a to b is A ෇ xab F͑x͒ dx, where F͑x͒ ജ 0. If the curve is given by parametric equations x ෇ f ͑t͒, y ෇ t͑t͒ and is traversed once as t increases from ␣ to ␤, then we can adapt the earlier formula by using the Substitution Rule for Definite Integrals as follows: b ␤ A ෇ y y dx ෇ y t͑t͒f Ј͑t͒ dt a ͫ or ␣ y␤ t͑t͒f Ј͑t͒ dt ␣ EXAMPLE 3 Find the area under one arch of the cycloid x ෇ r͑␪ Ϫ sin ␪ ͒, y ෇ r͑1 Ϫ cos ␪ ͒. (See Figure 3.) ͬ if ͑ f ͑ ␤͒, t͑ ␤͒͒ is the leftmost endpoint 5E-11(pp 696-705) 1/18/06 9:56 AM Page 699 SECTION 11.2 CALCULUS WITH PARAMETRIC CURVES ❙❙❙❙ 699 SOLUTION One arch of the cycloid is given by 0 ഛ ␪ ഛ 2␲. Using the Substitution Rule with y ෇ r͑1 Ϫ cos ␪͒ and dx ෇ r͑1 Ϫ cos ␪ ͒ d␪, we have y 0 A෇y x 2πr 2␲r y dx ෇ y FIGURE 3 ෇ r2 y 2␲ 0 |||| The result of Example 3 says that the area under one arch of the cycloid is three times the area of the rolling circle that generates the cycloid (see Example 6 in Section 11.1). Galileo guessed this result but it was first proved by the French mathematician Roberval and the Italian mathematician Torricelli. 2␲ 0 0 ෇ r2 y ͑1 Ϫ cos ␪ ͒2 d␪ ෇ r 2 y 2␲ ͑1 Ϫ 2 cos ␪ ϩ cos 2␪ ͒ d␪ 0 2␲ 0 r͑1 Ϫ cos ␪ ͒ r͑1 Ϫ cos ␪ ͒ d␪ [1 Ϫ 2 cos ␪ ϩ 1 2 ͑1 ϩ cos 2␪ ͒ d␪ [ ෇ r 2 3 ␪ Ϫ 2 sin ␪ ϩ 1 sin 2␪ 2 4 2␲ 0 ෇ r 2 ( 3 ؒ 2␲) ෇ 3␲ r 2 2 Arc Length We already know how to find the length L of a curve C given in the form y ෇ F͑x͒, a ഛ x ഛ b. Formula 9.1.3 says that if FЈ is continuous, then L෇ 3 yͱ ͩ ͪ b 2 dy dx 1ϩ a dx Suppose that C can also be described by the parametric equations x ෇ f ͑t͒ and y ෇ t͑t͒, ␣ ഛ t ഛ ␤, where dx͞dt ෇ f Ј͑t͒ Ͼ 0. This means that C is traversed once, from left to right, as t increases from ␣ to ␤ and f ͑␣͒ ෇ a, f ͑␤͒ ෇ b. Putting Formula 2 into Formula 3 and using the Substitution Rule, we obtain L෇ y b a y Pi _ 1 P™ Pi P¡ Pn P¸ FIGURE 4 1ϩ Since dx͞dt Ͼ 0, we have C 0 ͱ ͩ ͪ x 4 L෇ dy dx 2 dx ෇ y ␤ ␣ ͱ ͩ ͪ dy͞dt dx͞dt 1ϩ y ͱͩ ͪ ͩ ͪ ␤ ␣ 2 dx dt dy dt ϩ 2 dx dt dt 2 dt Even if C can’t be expressed in the form y ෇ F͑x͒, Formula 4 is still valid but we obtain it by polygonal approximations. We divide the parameter interval ͓␣, ␤͔ into n subintervals of equal width ⌬t. If t0 , t1 , t2 , . . . , tn are the endpoints of these subintervals, then xi ෇ f ͑ti ͒ and yi ෇ t͑ti ͒ are the coordinates of points Pi ͑xi , yi ͒ that lie on C and the polygon with vertices P0 , P1 , . . . , Pn approximates C (see Figure 4). As in Section 9.1, we define the length L of C to be the limit of the lengths of these approximating polygons as n l ϱ : n L ෇ lim ͚ ԽP n l ϱ i෇1 iϪ1 Pi Խ The Mean Value Theorem, when applied to f on the interval ͓tiϪ1, ti ͔, gives a number ti* in ͑tiϪ1, ti ͒ such that f ͑ti ͒ Ϫ f ͑tiϪ1 ͒ ෇ f Ј͑ti*͒͑ti Ϫ tiϪ1 ͒ If we let ⌬xi ෇ xi Ϫ xiϪ1 and ⌬yi ෇ yi Ϫ yiϪ1 , this equation becomes ⌬x i ෇ f Ј͑ti*͒ ⌬t 5E-11(pp 696-705) 700 ❙❙❙❙ 1/18/06 9:57 AM Page 700 CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES Similarly, when applied to t, the Mean Value Theorem gives a number ti** in ͑tiϪ1, ti ͒ such that ⌬yi ෇ tЈ͑ti**͒ ⌬t Therefore ԽP iϪ1 Խ Pi ෇ s͑⌬x i ͒2 ϩ ͑⌬yi ͒2 ෇ s͓ f Ј͑ti*͒⌬t͔ 2 ϩ ͓tЈ͑ti**͒⌬t͔ 2 ෇ s͓ f Ј͑ti*͔͒ 2 ϩ ͓tЈ͑ti**͔͒ 2 ⌬t and so n L ෇ lim 5 ͚ s͓ f Ј͑t*͔͒ i n l ϱ i෇1 2 ϩ ͓tЈ͑ti**͔͒ 2 ⌬t The sum in (5) resembles a Riemann sum for the function s͓ f Ј͑t͔͒ 2 ϩ ͓tЈ͑t͔͒ 2 but it is not exactly a Riemann sum because ti* ti** in general. Nevertheless, if f Ј and tЈ are continuous, it can be shown that the limit in (5) is the same as if ti* and ti** were equal, namely, ␤ L ෇ y s͓ f Ј͑t͔͒ 2 ϩ ͓tЈ͑t͔͒ 2 dt ␣ Thus, using Leibniz notation, we have the following result, which has the same form as (4). 6 Theorem If a curve C is described by the parametric equations x ෇ f ͑t͒, y ෇ t͑t͒, ␣ ഛ t ഛ ␤, where f Ј and tЈ are continuous on ͓␣, ␤͔ and C is traversed exactly once as t increases from ␣ to ␤, then the length of C is L෇ ␤ y ␣ ͱͩ ͪ ͩ ͪ 2 dx dt ϩ dy dt 2 dt Notice that the formula in Theorem 6 is consistent with the general formulas L ෇ x ds and ͑ds͒2 ෇ ͑dx͒2 ϩ ͑dy͒2 of Section 9.1. EXAMPLE 4 If we use the representation of the unit circle given in Example 2 in Section 11.1, x ෇ cos t y ෇ sin t 0 ഛ t ഛ 2␲ then dx͞dt ෇ Ϫsin t and dy͞dt ෇ cos t, so Theorem 6 gives L෇ y 2␲ 0 ͱͩ ͪ ͩ ͪ dx dt 2 ϩ dy dt 2 2␲ dt ෇ y ssin 2 t ϩ cos 2 t dt 0 2␲ ෇ y dt ෇ 2␲ 0 as expected. If, on the other hand, we use the representation given in Example 3 in Section 11.1, x ෇ sin 2t y ෇ cos 2t 0 ഛ t ഛ 2␲ 5E-11(pp 696-705) 1/18/06 9:58 AM Page 701 SECTION 11.2 CALCULUS WITH PARAMETRIC CURVES ❙❙❙❙ 701 then dx͞dt ෇ 2 cos 2t, dy͞dt ෇ Ϫ2 sin 2t, and the integral in Theorem 6 gives y 2␲ 0 ͱͩ ͪ ͩ ͪ dx dt 2 2 dy dt ϩ dt ෇ y 2␲ s4 cos 2 2t ϩ 4 sin 2 2t dt ෇ y 0 2␲ 0 2 dt ෇ 4␲ | Notice that the integral gives twice the arc length of the circle because as t increases from 0 to 2␲, the point ͑sin 2t, cos 2t͒ traverses the circle twice. In general, when finding the length of a curve C from a parametric representation, we have to be careful to ensure that C is traversed only once as t increases from ␣ to ␤. EXAMPLE 5 Find the length of one arch of the cycloid x ෇ r͑␪ Ϫ sin ␪ ͒, y ෇ r͑1 Ϫ cos ␪ ͒. SOLUTION From Example 3 we see that one arch is described by the parameter interval 0 ഛ ␪ ഛ 2␲. Since dx ෇ r͑1 Ϫ cos ␪ ͒ d␪ dy ෇ r sin ␪ d␪ and we have L෇ y 2␲ 0 |||| The result of Example 5 says that the length of one arch of a cycloid is eight times the radius of the generating circle (see Figure 5). This was first proved in 1658 by Sir Christopher Wren, who later became the architect of St. Paul’s Cathedral in London. y L=8r ෇y dx d␪ 2 dy d␪ ϩ 2 d␪ ෇ y 2␲ 0 sr 2͑1 Ϫ cos ␪ ͒2 ϩ r 2 sin 2␪ d␪ sr 2͑1 Ϫ 2 cos ␪ ϩ cos 2␪ ϩ sin 2␪ ͒ d␪ ෇ r y 2␲ 0 s2͑1 Ϫ cos ␪ ͒ d␪ To evaluate this integral we use the identity sin 2x ෇ 1 ͑1 Ϫ cos 2x͒ with ␪ ෇ 2x, which 2 gives 1 Ϫ cos ␪ ෇ 2 sin 2͑␪͞2͒. Since 0 ഛ ␪ ഛ 2␲, we have 0 ഛ ␪͞2 ഛ ␲ and so sin͑␪͞2͒ ജ 0. Therefore Խ Խ s2͑1 Ϫ cos ␪ ͒ ෇ s4 sin 2 ͑␪͞2͒ ෇ 2 sin͑␪͞2͒ ෇ 2 sin͑␪͞2͒ r 0 2␲ 0 ͱͩ ͪ ͩ ͪ 2πr x and so L ෇ 2r y 2␲ 0 sin͑␪͞2͒ d␪ ෇ 2r͓Ϫ2 cos͑␪͞2͒ 2␲ 0 ෇ 2r͓2 ϩ 2͔ ෇ 8r FIGURE 5 Surface Area In the same way as for arc length, we can adapt Formula 9.2.5 to obtain a formula for surface area. If the curve given by the parametric equations x ෇ f ͑t͒, y ෇ t͑t͒, ␣ ഛ t ഛ ␤, is rotated about the x-axis, where f Ј, tЈ are continuous and t͑t͒ ജ 0, then the area of the resulting surface is given by 7 ␤ ͱͩ ͪ ͩ ͪ dx dt S ෇ y 2␲ y ␣ 2 ϩ dy dt 2 dt The general symbolic formulas S ෇ x 2␲ y ds and S ෇ x 2␲ x ds (Formulas 9.2.7 and 9.2.8) are still valid, but for parametric curves we use ds ෇ ͱͩ ͪ ͩ ͪ dx dt 2 ϩ dy dt 2 dt 5E-11(pp 696-705) 702 ❙❙❙❙ 1/18/06 9:59 AM Page 702 CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES EXAMPLE 6 Show that the surface area of a sphere of radius r is 4␲ r 2. SOLUTION The sphere is obtained by rotating the semicircle x ෇ r cos t 0ഛtഛ␲ y ෇ r sin t about the x-axis. Therefore, from Formula 7, we get ␲ S ෇ y 2␲ r sin t s͑Ϫr sin t͒2 ϩ ͑r cos t͒2 dt 0 ␲ ෇ 2␲ y r sin t sr 2͑sin 2 t ϩ cos 2 t͒ dt 0 ␲ ␲ ෇ 2␲ y r sin t ؒ r dt ෇ 2␲r 2 y sin t dt 0 0 ␲ ෇ 2␲r 2͑Ϫcos t͒ 0 ෇ 4␲ r 2 |||| 11.2 1–2 Exercises 1. x ෇ t Ϫ t , 3 ■ ■ 3–6 17–20 |||| Find the points on the curve where the tangent is horizontal or vertical. If you have a graphing device, graph the curve to check your work. Find dy͞dx. |||| ■ y ෇ 2 Ϫ 5t ■ ■ 2. x ෇ te , t ■ ■ ■ y෇tϩe ■ ■ t ■ ■ 3. x ෇ t 4 ϩ 1, 5. x ෇ e , ■ 7–8 ■ ■ ■ ■ ■ ■ ■ ■ 8. x ෇ tan ␪, y ෇ sec ␪ ; 9–10 ■ 23. x ෇ t 4 Ϫ 2t 3 Ϫ 2t 2, ■ ■ ■ ■ ■ ■ Find an equation of the tangent(s) to the curve at the given ; point. Then graph the curve and the tangent(s). ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 2 2 |||| Find dy͞dx and d y͞dx . For which values of t is the curve concave upward? 11. x ෇ 4 ϩ t 2, y ෇ t2 ϩ t3 12. x ෇ t 3 Ϫ 12t, 13. x ෇ t Ϫ e t, y ෇ t ϩ e Ϫt 14. x ෇ t ϩ ln t, 15. x ෇ 2 sin t, y ෇ 3 cos t, ■ ■ ■ ■ 0 Ͻ t Ͻ 2␲ ■ ■ y ෇ t3 Ϫ t 24. x ෇ t 4 ϩ 4t 3 Ϫ 8t 2, ■ ■ ■ ■ y ෇ 2t 2 Ϫ t ■ ■ ■ ■ ■ ■ ■ ■ 25. Show that the curve x ෇ cos t, y ෇ sin t cos t has two tangents ■ y ෇ ͑tan t͒͑1 Ϫ 2 cos 2t͒ cross itself? Find the equations of both tangents at that point. 28. (a) Find the slope of the tangent to the astroid x ෇ a cos 3␪, 0ϽtϽ␲ ■ ■ x ෇ r␪ Ϫ d sin ␪, y ෇ r Ϫ d cos ␪ in terms of ␪. (See Exercise 38 in Section 11.1.) (b) Show that if d Ͻ r, then the trochoid does not have a vertical tangent. y ෇ t Ϫ ln t ■ ■ 27. (a) Find the slope of the tangent line to the trochoid y ෇ t2 Ϫ 1 y ෇ cos t , ■ 26. At what point does the curve x ෇ 1 Ϫ 2 cos 2t, ͑0, 0͒ 11–16 16. x ෇ cos 2t , ■ at ͑0, 0͒ and find their equations. Sketch the curve. (s3, 1) 10. x ෇ sin t, y ෇ sin͑t ϩ sin t͒; ■ |||| Graph the curve in a viewing rectangle that displays all the important aspects of the curve. |||| 9. x ෇ 2 sin 2t, y ෇ 2 sin t ; ■ most point on the curve x ෇ te t, y ෇ teϪt. Then find the exact coordinates. What are the asymptotes of this curve? (1, s2 ) ■ ■ ; 22. Try to estimate the coordinates of the highest point and the left- ͑1, 1͒ ■ ■ ; 23–24 7. x ෇ e t, y ෇ ͑t Ϫ 1͒ 2; ■ ■ the curve x ෇ t 4 Ϫ t 2, y ෇ t ϩ ln t. Then use calculus to find the exact coordinates. ■ Find an equation of the tangent to the curve at the given point by two methods: (a) without eliminating the parameter and (b) by first eliminating the parameter. ■ ■ ; 21. Use a graph to estimate the coordinates of the leftmost point on y ෇ sin ␪ ϩ cos 2␪ ; ␪ ෇ 0 |||| ■ y ෇ 2 sin ␪ ■ y ෇ t Ϫ ln t ; t ෇ 1 ■ y ෇ sin 2␪ 20. x ෇ cos 3␪, y ෇ 1 t 3 Ϫ t; t ෇ 3 3 6. x ෇ cos ␪ ϩ sin 2␪, y ෇ 2t 3 ϩ 3t 2 ϩ 1 19. x ෇ 2 cos ␪, 2 st ■ 18. x ෇ 2t 3 ϩ 3t 2 Ϫ 12t, y ෇ t 3 ϩ t ; t ෇ Ϫ1 4. x ෇ 2t 2 ϩ 1, y ෇ t 3 Ϫ 12t 17. x ෇ 10 Ϫ t 2, Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. |||| ■ ■ ■ ■ y ෇ a sin 3␪ in terms of ␪. (Astroids are explored in the Laboratory Project on page 695.) 5E-11(pp 696-705) 1/18/06 10:01 AM Page 703 SECTION 11.2 CALCULUS WITH PARAMETRIC CURVES 49. Use Simpson’s Rule with n ෇ 6 to estimate the length of the 29. At what points on the curve x ෇ t 3 ϩ 4t, y ෇ 6t 2 is the tangent curve x ෇ t Ϫ e t, y ෇ t ϩ e t, Ϫ6 ഛ t ഛ 6. parallel to the line with equations x ෇ Ϫ7t, y ෇ 12t Ϫ 5? 50. In Exercise 41 in Section 11.1 you were asked to derive the 30. Find equations of the tangents to the curve x ෇ 3t 2 ϩ 1, parametric equations x ෇ 2a cot ␪, y ෇ 2a sin 2␪ for the curve called the witch of Maria Agnesi. Use Simpson’s Rule with n ෇ 4 to estimate the length of the arc of this curve given by ␲͞4 ഛ ␪ ഛ ␲͞2. y ෇ 2t ϩ 1 that pass through the point ͑4, 3͒. 3 31. Use the parametric equations of an ellipse, x ෇ a cos ␪, y ෇ b sin ␪, 0 ഛ ␪ ഛ 2␲, to find the area that it encloses. 32. Find the area bounded by the curve x ෇ t Ϫ 1͞t, y ෇ t ϩ 1͞t 51–52 |||| Find the distance traveled by a particle with position ͑x, y͒ as t varies in the given time interval. Compare with the length of the curve. and the line y ෇ 2.5. 33. Find the area bounded by the curve x ෇ cos t, y ෇ e t, 51. x ෇ sin 2 t, 34. Find the area of the region enclosed by the astroid x ෇ a cos 3␪, y ෇ a sin ␪. (Astroids are explored in the Laboratory Project on page 695.) 3 y ෇ cos 2 t, 0 ഛ t ഛ 3␲ 52. x ෇ cos 2t, 0 ഛ t ഛ ␲͞2, and the lines y ෇ 1 and x ෇ 0. y ෇ cos t, ■ ■ ■ ■ L ෇ 4a y 36. Let ᏾ be the region enclosed by the loop of the curve in 40. x ෇ ln t, ■ ■ CAS ■ ■ ■ s1 Ϫ e 2 sin 2␪ d␪ y ෇ 11 sin t Ϫ 4 sin͑11t͞2͒ 0 ഛ t ഛ 2␲ ■ ■ x ෇ 11 cos t Ϫ 4 cos͑11t͞2͒ Ϫ3 ഛ t ഛ 3 ■ ■ 55. (a) Graph the epitrochoid with equations What parameter interval gives the complete curve? (b) Use your CAS to find the approximate length of this curve. y ෇ st ϩ 1, 1 ഛ t ഛ 5 ■ ■ where a Ͼ 0. 1ഛtഛ2 y ෇ t Ϫ sin t, ■ 54. Find the total length of the astroid x ෇ a cos 3␪, y ෇ a sin 3␪, Set up, but do not evaluate, an integral that represents the length of the curve. 39. x ෇ t ϩ cos t, ■ where e is the eccentricity of the ellipse (e ෇ c͞a, where c ෇ sa 2 Ϫ b 2 ) . |||| y ෇ t 2, ␲͞2 0 Example 1. (a) Find the area of ᏾. (b) If ᏾ is rotated about the x-axis, find the volume of the resulting solid. (c) Find the centroid of ᏾. 38. x ෇ 1 ϩ e t, ■ y ෇ b cos ␪, a Ͼ b Ͼ 0, is Section 11.1 for the case d Ͻ r. y ෇ 4 t 3͞2, 3 0 ഛ t ഛ 4␲ ■ 53. Show that the total length of the ellipse x ෇ a sin ␪, 35. Find the area under one arch of the trochoid of Exercise 38 in 37. x ෇ t Ϫ t 2, 703 48. Find the length of the loop of the curve x ෇ 3t Ϫ t 3, y ෇ 3t 2. (b) At what points is the tangent horizontal or vertical? (c) At what points does the tangent have slope 1 or Ϫ1? 37–40 ❙❙❙❙ ■ ■ ■ ■ ■ CAS 56. A curve called Cornu’s spiral is defined by the parametric equations 41–44 |||| Find the length of the curve. 41. x ෇ 1 ϩ 3t , y ෇ 4 ϩ 2t , 2 3 42. x ෇ a͑cos ␪ ϩ ␪ sin ␪ ͒, 0ഛ␪ഛ␲ 43. x ෇ t , 1ϩt ■ ; 45–47 ■ |||| ■ ■ 0 ■ ■ ■ ■ ■ 57–58 |||| Set up, but do not evaluate, an integral that represents the area of the surface obtained by rotating the given curve about the x-axis. y ෇ e t sin t, 0 ഛ t ഛ ␲ 1 47. x ෇ e Ϫ t, t ■ where C and S are the Fresnel functions that were introduced in Chapter 5. (a) Graph this curve. What happens as t l ϱ and as t l Ϫϱ? (b) Find the length of Cornu’s spiral from the origin to the point with parameter value t. ■ Graph the curve and find its length. 46. x ෇ cos t ϩ ln(tan 2 t), ■ t y ෇ S͑t͒ ෇ y sin͑␲ u 2͞2͒ du y ෇ a͑sin ␪ Ϫ ␪ cos ␪ ͒, y ෇ 5 Ϫ 2t, 0 ഛ t ഛ 3 ■ 45. x ෇ e t cos t, ■ 0ഛtഛ1 0 y ෇ ln͑1 ϩ t͒, 0 ഛ t ഛ 2 44. x ෇ e t ϩ eϪt, ■ t x ෇ C͑t͒ ෇ y cos͑␲ u 2͞2͒ du y ෇ 4e , t͞2 ■ ■ y ෇ sin t, ␲͞4 ഛ t ഛ 3␲͞4 57. x ෇ t Ϫ t 2, 58. x ෇ sin t, Ϫ8 ഛ t ഛ 3 ■ ■ ■ y ෇ 4 t 3͞2, 3 y ෇ sin 3t, 2 ■ ■ ■ ■ ■ ■ ■ ■ ■ 1ഛtഛ2 0 ഛ t ഛ ␲͞3 ■ ■ ■ ■ ■ ■ ■ 5E-11(pp 696-705) 704 ❙❙❙❙ 1/18/06 10:01 AM Page 704 CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES 59–61 |||| Find the area of the surface obtained by rotating the given curve about the x-axis. 59. x ෇ t 3, y ෇ t 2, 60. x ෇ 3t Ϫ t , 61. x ෇ a cos 3␪, ■ ■ ■ 0ഛtഛ1 y ෇ 3t 2, 3 0ഛtഛ1 ␬෇ y ෇ a sin 3␪, 0 ഛ ␪ ഛ ␲͞2 ■ ■ ■ (b) By regarding a curve y ෇ f ͑x͒ as the parametric curve x ෇ x, y ෇ f ͑x͒, with parameter x, show that the formula in part (a) becomes ■ ■ ■ ■ ■ ■ Խd Խ 2 y͞dx 2 ͓1 ϩ ͑dy͞dx͒2 ͔ 3͞2 y ; 62. Graph the curve x ෇ 2 cos ␪ Ϫ cos 2␪ P y ෇ 2 sin ␪ Ϫ sin 2␪ If this curve is rotated about the x-axis, find the area of the resulting surface. (Use your graph to help find the correct parameter interval.) ˙ 0 63. If the curve x ෇ t ϩ t3 y෇tϪ 70. (a) Use the formula in Exercise 69(b) to find the curvature of 1 t2 the parabola y ෇ x 2 at the point ͑1, 1͒. (b) At what point does this parabola have maximum curvature? 1ഛtഛ2 is rotated about the x-axis, use your calculator to estimate the area of the resulting surface to three decimal places. 71. Use the formula in Exercise 69(a) to find the curvature of the cycloid x ෇ ␪ Ϫ sin ␪, y ෇ 1 Ϫ cos ␪ at the top of one of its arches. 64. If the arc of the curve in Exercise 50 is rotated about the x-axis, estimate the area of the resulting surface using Simpson’s Rule with n ෇ 4. 72. (a) Show that the curvature at each point of a straight line is ␬ ෇ 0. (b) Show that the curvature at each point of a circle of radius r is ␬ ෇ 1͞r. 65–66 |||| Find the surface area generated by rotating the given curve about the y-axis. 65. x ෇ 3t 2, y ෇ 2t 3, 66. x ෇ e t Ϫ t, ■ ■ ■ ■ 73. A string is wound around a circle and then unwound while 0ഛtഛ5 y ෇ 4e t͞2, 0ഛtഛ1 ■ ■ x ■ ■ ■ ■ ■ ■ 67. If f Ј is continuous and f Ј͑t͒ 0 for a ഛ t ഛ b, show that the parametric curve x ෇ f ͑t͒, y ෇ t͑t͒, a ഛ t ഛ b, can be put in the form y ෇ F͑x͒. [Hint: Show that f Ϫ1 exists.] 68. Use Formula 2 to derive Formula 7 from Formula 9.2.5 for the being held taut. The curve traced by the point P at the end of the string is called the involute of the circle. If the circle has radius r and center O and the initial position of P is ͑r, 0͒, and if the parameter ␪ is chosen as in the figure, show that parametric equations of the involute are x ෇ r͑cos ␪ ϩ ␪ sin ␪ ͒ y ෇ r͑sin ␪ Ϫ ␪ cos ␪ ͒ y case in which the curve can be represented in the form y ෇ F͑x͒, a ഛ x ഛ b. T r 69. The curvature at a point P of a curve is defined as ␬෇ Ϳ Ϳ O d␾ ds where ␾ is the angle of inclination of the tangent line at P, as shown in the figure. Thus, the curvature is the absolute value of the rate of change of ␾ with respect to arc length. It can be regarded as a measure of the rate of change of direction of the curve at P and will be studied in greater detail in Chapter 14. (a) For a parametric curve x ෇ x͑t͒, y ෇ y͑t͒, derive the formula ␬෇ Խ ¨ Խ x ᝽᝽ Ϫ ᝽᝽y ᝽y x᝽ ͓x 2 ϩ y 2 ͔ 3͞2 ᝽ ᝽ where the dots indicate derivatives with respect to t, so x ෇ dx͞dt. [Hint: Use ␾ ෇ tanϪ1͑dy͞dx͒ and Equation 2 ᝽ to find d␾͞dt. Then use the Chain Rule to find d␾͞ds.] P x 74. A cow is tied to a silo with radius r by a rope just long enough to reach the opposite side of the silo. Find the area available for grazing by the cow. 5E-11(pp 696-705) 1/18/06 10:02 AM Page 705 SECTION 11.3 POLAR COORDINATES ❙❙❙❙ 705 LABORATORY PROJECT ; Bézier Curves The Bézier curves are used in computer-aided design and are named after the French mathematician Pierre Bézier (1910–1999), who worked in the automotive industry. A cubic Bézier curve is determined by four control points, P0͑x 0 , y0 ͒, P1͑x 1, y1 ͒, P2͑x 2 , y2 ͒, and P3͑x 3 , y3 ͒, and is defined by the parametric equations x ෇ x0͑1 Ϫ t͒3 ϩ 3x1 t͑1 Ϫ t͒2 ϩ 3x 2 t 2͑1 Ϫ t͒ ϩ x 3 t 3 y ෇ y0͑1 Ϫ t͒3 ϩ 3y1 t͑1 Ϫ t͒2 ϩ 3y2 t 2͑1 Ϫ t͒ ϩ y3 t 3 where 0 ഛ t ഛ 1. Notice that when t ෇ 0 we have ͑x, y͒ ෇ ͑x 0 , y0 ͒ and when t ෇ 1 we have ͑x, y͒ ෇ ͑x 3 , y3 ͒, so the curve starts at P0 and ends at P3 . 1. Graph the Bézier curve with control points P0͑4, 1͒, P1͑28, 48͒, P2͑50, 42͒, and P3͑40, 5͒. Then, on the same screen, graph the line segments P0 P1, P1 P2, and P2 P3. (Exercise 31 in Section 11.1 shows how to do this.) Notice that the middle control points P1 and P2 don’t lie on the curve; the curve starts at P0, heads toward P1 and P2 without reaching them, and ends at P3. 2. From the graph in Problem 1 it appears that the tangent at P0 passes through P1 and the tangent at P3 passes through P2. Prove it. 3. Try to produce a Bézier curve with a loop by changing the second control point in Problem 1. 4. Some laser printers use Bézier curves to represent letters and other symbols. Experiment with control points until you find a Bézier curve that gives a reasonable representation of the letter C. 5. More complicated shapes can be represented by piecing together two or more Bézier curves. Suppose the first Bézier curve has control points P0 , P1, P2 , P3 and the second one has control points P3 , P4 , P5 , P6. If we want these two pieces to join together smoothly, then the tangents at P3 should match and so the points P2, P3, and P4 all have to lie on this common tangent line. Using this principle, find control points for a pair of Bézier curves that represent the letter S. |||| 11.3 Polar Coordinates P (r, ¨ ) r O ¨ FIGURE 1 polar axis x A coordinate system represents a point in the plane by an ordered pair of numbers called coordinates. Usually we use Cartesian coordinates, which are directed distances from two perpendicular axes. Here we describe a coordinate system introduced by Newton, called the polar coordinate system, which is more convenient for many purposes. We choose a point in the plane that is called the pole (or origin) and is labeled O. Then we draw a ray (half-line) starting at O called the polar axis. This axis is usually drawn horizontally to the right and corresponds to the positive x-axis in Cartesian coordinates. If P is any other point in the plane, let r be the distance from O to P and let ␪ be the angle (usually measured in radians) between the polar axis and the line OP as in Figure 1. Then the point P is represented by the ordered pair ͑r, ␪͒ and r, ␪ are called polar coordinates of P. We use the convention that an angle is positive if measured in the counterclockwise direction from the polar axis and negative in the clockwise direction. If P ෇ O, then r ෇ 0 and we agree that ͑0, ␪͒ represents the pole for any value of ␪. 5E-11(pp 706-715) 706 ❙❙❙❙ 1/18/06 9:48 AM Page 706 CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES We extend the meaning of polar coordinates ͑r, ␪ ͒ to the case in which r is negative by agreeing that, as in Figure 2, the points ͑Ϫr, ␪ ͒ and ͑r, ␪ ͒ lie on the same line through O and at the same distance r from O, but on opposite sides of O. If r Ͼ 0, the point ͑r, ␪ ͒ lies in the same quadrant as ␪ ; if r Ͻ 0, it lies in the quadrant on the opposite side of the pole. Notice that ͑Ϫr, ␪ ͒ represents the same point as ͑r, ␪ ϩ ␲͒. (r, ¨ ) ¨+π Խ Խ ¨ O EXAMPLE 1 Plot the points whose polar coordinates are given. (a) ͑1, 5␲͞4͒ (b) ͑2, 3␲͒ (c) ͑2, Ϫ2␲͞3͒ (d) ͑Ϫ3, 3␲͞4͒ (_r, ¨) FIGURE 2 5π 4 SOLUTION The points are plotted in Figure 3. In part (d) the point ͑Ϫ3, 3␲͞4͒ is located three units from the pole in the fourth quadrant because the angle 3␲͞4 is in the second quadrant and r ෇ Ϫ3 is negative. 3π O (2, 3π) 3π 4 O O O _ 2π 3 5π ”1, ’ 4 2π ”2, _ ’ 3 FIGURE 3 ”_3, 3π ’ 4 In the Cartesian coordinate system every point has only one representation, but in the polar coordinate system each point has many representations. For instance, the point ͑1, 5␲͞4͒ in Example 1(a) could be written as ͑1, Ϫ3␲͞4͒ or ͑1, 13␲͞4͒ or ͑Ϫ1, ␲͞4͒. (See Figure 4.) 5π 4 13π 4 O O _ 3π 4 ”1, 5π ’ 4 3π ”1, _ 4 ’ π 4 O O 13π ”1, 4 ’ π ”_1, ’ 4 FIGURE 4 In fact, since a complete counterclockwise rotation is given by an angle 2␲, the point represented by polar coordinates ͑r, ␪ ͒ is also represented by ͑r, ␪ ϩ 2n␲͒ y P (r, ¨ )=P (x, y) r y ¨ O FIGURE 5 x x and ͑Ϫr, ␪ ϩ ͑2n ϩ 1͒␲͒ where n is any integer. The connection between polar and Cartesian coordinates can be seen from Figure 5, in which the pole corresponds to the origin and the polar axis coincides with the positive x-axis. If the point P has Cartesian coordinates ͑x, y͒ and polar coordinates ͑r, ␪ ͒, then, from the figure, we have x y cos ␪ ෇ sin ␪ ෇ r r and so 1 x ෇ r cos ␪ y ෇ r sin ␪ Although Equations 1 were deduced from Figure 5, which illustrates the case where r Ͼ 0 and 0 Ͻ ␪ Ͻ ␲͞2, these equations are valid for all values of r and ␪. (See the general definition of sin ␪ and cos ␪ in Appendix D.) 5E-11(pp 706-715) 1/18/06 9:48 AM Page 707 SECTION 11.3 POLAR COORDINATES ❙❙❙❙ 707 Equations 1 allow us to find the Cartesian coordinates of a point when the polar coordinates are known. To find r and ␪ when x and y are known, we use the equations r2 ෇ x2 ϩ y2 2 tan ␪ ෇ y x which can be deduced from Equations 1 or simply read from Figure 5. EXAMPLE 2 Convert the point ͑2, ␲͞3͒ from polar to Cartesian coordinates. SOLUTION Since r ෇ 2 and ␪ ෇ ␲͞3, Equations 1 give x ෇ r cos ␪ ෇ 2 cos y ෇ r sin ␪ ෇ 2 sin ␲ 1 ෇2ؒ ෇1 3 2 ␲ s3 ෇2ؒ ෇ s3 3 2 Therefore, the point is (1, s3 ) in Cartesian coordinates. EXAMPLE 3 Represent the point with Cartesian coordinates ͑1, Ϫ1͒ in terms of polar coordinates. SOLUTION If we choose r to be positive, then Equations 2 give r ෇ sx 2 ϩ y 2 ෇ s1 2 ϩ ͑Ϫ1͒ 2 ෇ s2 tan ␪ ෇ y ෇ Ϫ1 x Since the point ͑1, Ϫ1͒ lies in the fourth quadrant, we can choose ␪ ෇ Ϫ␲͞4 or ␪ ෇ 7␲͞4. Thus, one possible answer is (s2, Ϫ␲͞4); another is ͑s2, 7␲͞4͒. NOTE Equations 2 do not uniquely determine ␪ when x and y are given because, as ␪ increases through the interval 0 ഛ ␪ Ͻ 2␲, each value of tan ␪ occurs twice. Therefore, in converting from Cartesian to polar coordinates, it’s not good enough just to find r and ␪ that satisfy Equations 2. As in Example 3, we must choose ␪ so that the point ͑r, ␪ ͒ lies in the correct quadrant. ■ Polar Curves 1 r= 2 r=4 The graph of a polar equation r ෇ f ͑␪ ͒, or more generally F͑r, ␪ ͒ ෇ 0, consists of all points P that have at least one polar representation ͑r, ␪ ͒ whose coordinates satisfy the equation. r=2 r=1 x FIGURE 6 EXAMPLE 4 What curve is represented by the polar equation r ෇ 2? SOLUTION The curve consists of all points ͑r, ␪ ͒ with r ෇ 2. Since r represents the distance from the point to the pole, the curve r ෇ 2 represents the circle with center O and radius 2. In general, the equation r ෇ a represents a circle with center O and radius a . (See Figure 6.) Խ Խ 5E-11(pp 706-715) 708 ❙❙❙❙ 1/18/06 9:48 AM Page 708 CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES EXAMPLE 5 Sketch the polar curve ␪ ෇ 1. (3, 1) SOLUTION This curve consists of all points ͑r, ␪ ͒ such that the polar angle ␪ is 1 radian. It is the straight line that passes through O and makes an angle of 1 radian with the polar axis (see Figure 7). Notice that the points ͑r, 1͒ on the line with r Ͼ 0 are in the first quadrant, whereas those with r Ͻ 0 are in the third quadrant. (2, 1) ¨=1 (1, 1) O 1 x EXAMPLE 6 (_1, 1) (a) Sketch the curve with polar equation r ෇ 2 cos ␪. (b) Find a Cartesian equation for this curve. (_2, 1) SOLUTION FIGURE 7 (a) In Figure 8 we find the values of r for some convenient values of ␪ and plot the corresponding points ͑r, ␪ ͒. Then we join these points to sketch the curve, which appears to be a circle. We have used only values of ␪ between 0 and ␲, since if we let ␪ increase beyond ␲, we obtain the same points again. ␪ FIGURE 8 Table of values and graph of r=2 cos ¨ r ෇ 2 cos ␪ 0 ␲͞6 ␲͞4 ␲͞3 ␲͞2 2␲͞3 3␲͞4 5␲͞6 ␲ 2 s3 s2 1 0 Ϫ1 Ϫs2 Ϫs3 Ϫ2 π ”1, ’ 3 ” œ„, ’ 2 π 4 ”œ„, ’ 3 π 6 (2, 0) π ”0, ’ 2 2π ”_1, ’ 3 ”_ œ„, ’ 2 3π 4 ’ ”_ œ„, 5π 3 6 (b) To convert the given equation into a Cartesian equation we use Equations 1 and 2. From x ෇ r cos ␪ we have cos ␪ ෇ x͞r, so the equation r ෇ 2 cos ␪ becomes r ෇ 2x͞r, which gives 2x ෇ r 2 ෇ x 2 ϩ y 2 or x 2 ϩ y 2 Ϫ 2x ෇ 0 Completing the square, we obtain ͑x Ϫ 1͒2 ϩ y 2 ෇ 1 which is an equation of a circle with center ͑1, 0͒ and radius 1. |||| Figure 9 shows a geometrical illustration that the circle in Example 6 has the equation r ෇ 2 cos ␪. The angle OPQ is a right angle (Why?) and so r͞2 ෇ cos ␪. y P r ¨ O FIGURE 9 2 Q x 5E-11(pp 706-715) 1/18/06 9:48 AM Page 709 SECTION 11.3 POLAR COORDINATES r ❙❙❙❙ 709 EXAMPLE 7 Sketch the curve r ෇ 1 ϩ sin ␪. 2 SOLUTION Instead of plotting points as in Example 6, we first sketch the graph of r ෇ 1 ϩ sin ␪ in Cartesian coordinates in Figure 10 by shifting the sine curve up one unit. This enables us to read at a glance the values of r that correspond to increasing values of ␪. For instance, we see that as ␪ increases from 0 to ␲͞2, r (the distance from O ) increases from 1 to 2, so we sketch the corresponding part of the polar curve in Figure 11(a). As ␪ increases from ␲͞2 to ␲, Figure 10 shows that r decreases from 2 to 1, so we sketch the next part of the curve as in Figure 11(b). As ␪ increases from ␲ to 3␲͞2, r decreases from 1 to 0 as shown in part (c). Finally, as ␪ increases from 3␲͞2 to 2␲, r increases from 0 to 1 as shown in part (d). If we let ␪ increase beyond 2␲ or decrease beyond 0, we would simply retrace our path. Putting together the parts of the curve from Figure 11(a)–(d), we sketch the complete curve in part (e). It is called a cardioid because it’s shaped like a heart. 1 0 π π 2 2π ¨ 3π 2 FIGURE 10 r=1+sin ¨ in Cartesian coordinates, 0¯¨¯2π π π ¨= 2 ¨= 2 2 O O 1 O ¨=0 ¨=π O ¨=π 3π O 3π ¨= 2 (a) ¨= 2 (c) (b) ¨=2π (d) (e) FIGURE 11 Stages in sketching the cardioid r=1+sin ¨ EXAMPLE 8 Sketch the curve r ෇ cos 2␪. Module 11.3 helps you see how polar curves are traced out by showing animations similar to Figures 10–13. Tangents to these polar curves can also be visualized as in Figure 15 (see page 712). SOLUTION As in Example 7, we first sketch r ෇ cos 2␪, 0 ഛ ␪ ഛ 2␲, in Cartesian coordinates in Figure 12. As ␪ increases from 0 to ␲͞4, Figure 12 shows that r decreases from 1 to 0 and so we draw the corresponding portion of the polar curve in Figure 13 (indicated by !). As ␪ increases from ␲͞4 to ␲͞2, r goes from 0 to Ϫ1. This means that the distance from O increases from 0 to 1, but instead of being in the first quadrant this portion of the polar curve (indicated by @) lies on the opposite side of the pole in the third quadrant. The remainder of the curve is drawn in a similar fashion, with the arrows and numbers indicating the order in which the portions are traced out. The resulting curve has four loops and is called a four-leaved rose. r π ¨= 2 1 ¨= $ ! π 4 @ π 2 3π 4 # % π * 5π 4 3π 2 ^ 7π 4 π 3π 4 & ¨= 4 ^ $ 2π ¨ ! % ⑧ ¨=π & ¨=0 @ # FIGURE 12 FIGURE 13 r=cos 2¨ in Cartesian coordinates Four-leaved rose r=cos 2¨ 5E-11(pp 706-715) 710 ❙❙❙❙ 1/18/06 9:49 AM Page 710 CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES Symmetry When we sketch polar curves it is sometimes helpful to take advantage of symmetry. The following three rules are explained by Figure 14. (a) If a polar equation is unchanged when ␪ is replaced by Ϫ␪, the curve is symmetric about the polar axis. (b) If the equation is unchanged when r is replaced by Ϫr, or when ␪ is replaced by ␪ ϩ ␲, the curve is symmetric about the pole. (This means that the curve remains unchanged if we rotate it through 180° about the origin.) (c) If the equation is unchanged when ␪ is replaced by ␲ Ϫ ␪, the curve is symmetric about the vertical line ␪ ෇ ␲͞2. (r, π-¨ ) (r, ¨ ) (r, ¨ ) (r, ¨) π-¨ ¨ O ¨ (_ r, ¨ ) _¨ O O (r, _¨ ) (a) (b) (c) FIGURE 14 The curves sketched in Examples 6 and 8 are symmetric about the polar axis, since cos͑Ϫ␪ ͒ ෇ cos ␪. The curves in Examples 7 and 8 are symmetric about ␪ ෇ ␲͞2 because sin͑␲ Ϫ ␪ ͒ ෇ sin ␪ and cos 2͑␲ Ϫ ␪ ͒ ෇ cos 2␪. The four-leaved rose is also symmetric about the pole. These symmetry properties could have been used in sketching the curves. For instance, in Example 6 we need only have plotted points for 0 ഛ ␪ ഛ ␲͞2 and then reflected about the polar axis to obtain the complete circle. Tangents to Polar Curves To find a tangent line to a polar curve r ෇ f ͑␪ ͒ we regard ␪ as a parameter and write its parametric equations as x ෇ r cos ␪ ෇ f ͑␪ ͒ cos ␪ y ෇ r sin ␪ ෇ f ͑␪ ͒ sin ␪ Then, using the method for finding slopes of parametric curves (Equation 11.2.2) and the Product Rule, we have dy dr sin ␪ ϩ r cos ␪ dy d␪ d␪ ෇ ෇ 3 dx dx dr cos ␪ Ϫ r sin ␪ d␪ d␪ We locate horizontal tangents by finding the points where dy͞d␪ ෇ 0 (provided that dx͞d␪ 0 ). Likewise, we locate vertical tangents at the points where dx͞d␪ ෇ 0 (provided that dy͞d␪ 0). Notice that if we are looking for tangent lines at the pole, then r ෇ 0 and Equation 3 simplifies to dy dr ෇ tan ␪ if 0 dx d␪ 5E-11(pp 706-715) 1/18/06 9:50 AM Page 711 SECTION 11.3 POLAR COORDINATES ❙❙❙❙ 711 For instance, in Example 8 we found that r ෇ cos 2␪ ෇ 0 when ␪ ෇ ␲͞4 or 3␲͞4. This means that the lines ␪ ෇ ␲͞4 and ␪ ෇ 3␲͞4 (or y ෇ x and y ෇ Ϫx) are tangent lines to r ෇ cos 2␪ at the origin. EXAMPLE 9 (a) For the cardioid r ෇ 1 ϩ sin ␪ of Example 7, find the slope of the tangent line when ␪ ෇ ␲͞3. (b) Find the points on the cardioid where the tangent line is horizontal or vertical. SOLUTION Using Equation 3 with r ෇ 1 ϩ sin ␪, we have dr sin ␪ ϩ r cos ␪ dy d␪ cos ␪ sin ␪ ϩ ͑1 ϩ sin ␪ ͒ cos ␪ ෇ ෇ dx dr cos ␪ cos ␪ Ϫ ͑1 ϩ sin ␪ ͒ sin ␪ cos ␪ Ϫ r sin ␪ d␪ ෇ cos ␪ ͑1 ϩ 2 sin ␪ ͒ cos ␪ ͑1 ϩ 2 sin ␪ ͒ ෇ 2 1 Ϫ 2 sin ␪ Ϫ sin ␪ ͑1 ϩ sin ␪ ͒͑1 Ϫ 2 sin ␪ ͒ (a) The slope of the tangent at the point where ␪ ෇ ␲͞3 is Ϳ cos͑␲͞3͒͑1 ϩ 2 sin͑␲͞3͒͒ ͑1 ϩ sin͑␲͞3͒͒͑1 Ϫ 2 sin͑␲͞3͒͒ (1 ϩ s3 ) (1 ϩ s3͞2)(1 Ϫ s3 ) ෇ ␪ ෇␲ ͞3 ෇ ෇ dy dx 1 ϩ s3 ෇ Ϫ1 Ϫ1 Ϫ s3 1 2 ෇ 1 ϩ s3 (2 ϩ s3 )(1 Ϫ s3 ) (b) Observe that dy ෇ cos ␪ ͑1 ϩ 2 sin ␪ ͒ ෇ 0 d␪ when ␪ ෇ ␲ 3␲ 7␲ 11␲ , , , 2 2 6 6 dx ෇ ͑1 ϩ sin ␪ ͒͑1 Ϫ 2 sin ␪ ͒ ෇ 0 d␪ when ␪ ෇ 3␲ ␲ 5␲ , , 2 6 6 Therefore, there are horizontal tangents at the points ͑2, ␲͞2͒, ( 1 , 7␲͞6), ( 1 , 11␲͞6) and 2 2 vertical tangents at ( 3 , ␲͞6) and ( 3 , 5␲͞6). When ␪ ෇ 3␲͞2, both dy͞d␪ and dx͞d␪ are 2 2 0, so we must be careful. Using l’Hospital’s Rule, we have lim ␪ l ͑3␲ ͞2͒Ϫ dy ෇ dx ͩ lim ␪ l ͑3␲ ͞2͒Ϫ 1 ϩ 2 sin ␪ 1 Ϫ 2 sin ␪ ͪͩ lim cos ␪ 1 ϩ sin ␪ lim Ϫsin ␪ ෇ϱ cos ␪ ෇Ϫ 1 3 ␪ l ͑3␲ ͞2͒Ϫ ෇Ϫ 1 3 ␪ l ͑3␲ ͞2͒Ϫ lim ␪ l ͑3␲ ͞2͒Ϫ cos ␪ 1 ϩ sin ␪ ͪ 5E-11(pp 706-715) ❙❙❙❙ 712 1/18/06 9:50 AM Page 712 CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES π By symmetry, ”2, ’ 2 3 ”1+ œ„ , ’ π 2 3 m=_1 dy ෇ Ϫϱ dx Thus, there is a vertical tangent line at the pole (see Figure 15). NOTE Instead of having to remember Equation 3, we could employ the method used to derive it. For instance, in Example 9 we could have written ”3 π , ’ 2 6 3 ” , 5π ’ 2 6 lim ␪ l ͑3␲͞2͒ϩ ■ (0, 0) x ෇ r cos ␪ ෇ ͑1 ϩ sin ␪ ͒ cos ␪ ෇ cos ␪ ϩ 1 sin 2␪ 2 1 1 ” , 7π ” , 11π ’ 2 ’ 2 6 6 y ෇ r sin ␪ ෇ ͑1 ϩ sin ␪ ͒ sin ␪ ෇ sin ␪ ϩ sin 2␪ FIGURE 15 Then we have Tangent lines for r=1+sin ¨ dy dy͞d␪ cos ␪ ϩ 2 sin ␪ cos ␪ cos ␪ ϩ sin 2␪ ෇ ෇ ෇ dx dx͞d␪ Ϫsin ␪ ϩ cos 2␪ Ϫsin ␪ ϩ cos 2␪ which is equivalent to our previous expression. Graphing Polar Curves with Graphing Devices 2 _1.4 1.4 _0.3 Although it’s useful to be able to sketch simple polar curves by hand, we need to use a graphing calculator or computer when we are faced with a curve as complicated as the one shown in Figure 16. Some graphing devices have commands that enable us to graph polar curves directly. With other machines we need to convert to parametric equations first. In this case we take the polar equation r ෇ f ͑␪ ͒ and write its parametric equations as x ෇ r cos ␪ ෇ f ͑␪ ͒ cos ␪ FIGURE 16 y ෇ r sin ␪ ෇ f ͑␪ ͒ sin ␪ Some machines require that the parameter be called t rather than ␪. r=sin ¨+sin#(5¨/2) EXAMPLE 10 Graph the curve r ෇ sin͑8␪͞5͒. SOLUTION Let’s assume that our graphing device doesn’t have a built-in polar graphing command. In this case we need to work with the corresponding parametric equations, which are 1 _1 1 _1 FIGURE 17 r=sin(8¨/5) x ෇ r cos ␪ ෇ sin͑8␪͞5͒ cos ␪ y ෇ r sin ␪ ෇ sin͑8␪͞5͒ sin ␪ In any case we need to determine the domain for ␪. So we ask ourselves: How many complete rotations are required until the curve starts to repeat itself? If the answer is n, then 8␪ 8␪ 8͑␪ ϩ 2n␲͒ 16n␲ ෇ sin sin ෇ sin ϩ 5 5 5 5 ͩ ͪ and so we require that 16n␲͞5 be an even multiple of ␲. This will first occur when n ෇ 5. Therefore, we will graph the entire curve if we specify that 0 ഛ ␪ ഛ 10␲. Switching from ␪ to t, we have the equations x ෇ sin͑8t͞5͒ cos t y ෇ sin͑8t͞5͒ sin t 0 ഛ t ഛ 10␲ and Figure 17 shows the resulting curve. Notice that this rose has 16 loops. 5E-11(pp 706-715) 1/18/06 9:51 AM Page 713 SECTION 11.3 POLAR COORDINATES ❙❙❙❙ 713 EXAMPLE 11 Investigate the family of polar curves given by r ෇ 1 ϩ c sin ␪. How does the shape change as c changes? (These curves are called limaçons, after a French word for snail, because of the shape of the curves for certain values of c.) SOLUTION Figure 18 shows computer-drawn graphs for various values of c. For c Ͼ 1 there is a loop that decreases in size as c decreases. When c ෇ 1 the loop disappears and the curve becomes the cardioid that we sketched in Example 7. For c between 1 and 1 the 2 cardioid’s cusp is smoothed out and becomes a “dimple.” When c decreases from 1 to 0, 2 the limaçon is shaped like an oval. This oval becomes more circular as c l 0, and when c ෇ 0 the curve is just the circle r ෇ 1. |||| In Exercise 53 you are asked to prove analytically what we have discovered from the graphs in Figure 18. c=1 c=1.7 c=0.7 c=0.5 c=0.2 c=2.5 c=_2 c=0 FIGURE 18 c=_ 0.8 Exercises 6. (a) (Ϫ1, Ϫs3 ) 1–2 |||| Plot the point whose polar coordinates are given. Then find two other pairs of polar coordinates of this point, one with r Ͼ 0 and one with r Ͻ 0. 1. (a) ͑1, ␲͞2͒ (b) ͑Ϫ2, ␲͞4͒ ■ 3–4 ■ ■ ■ ■ ■ (c) ͑Ϫ1, Ϫ␲͞2͒ ■ ■ ■ ■ ■ 4. (a) ͑2, 2␲͞3͒ ■ ■ 5–6 ■ ■ ■ ■ ■ ■ ■ The Cartesian coordinates of a point are given. (i) Find polar coordinates ͑r, ␪ ͒ of the point, where r Ͼ 0 and 0 ഛ ␪ Ͻ 2␲. (ii) Find polar coordinates ͑r, ␪ ͒ of the point, where r Ͻ 0 and 0 ഛ ␪ Ͻ 2␲. |||| 5. (a) ͑1, 1͒ (b) (2 s3, Ϫ2) ■ ■ ■ 9. 0 ഛ r Ͻ 4, ■ ■ ■ ■ ■ Ϫ␲͞2 ഛ ␪ Ͻ ␲͞6 3␲͞4 Ͻ ␪ Ͻ 5␲͞4 11. 2 Ͻ r Ͻ 3, (c) ͑Ϫ2, Ϫ5␲͞6͒ ■ (b) ͑Ϫ2, 3͒ ■ 10. 2 Ͻ r ഛ 5, (c) ͑Ϫ1, ␲͞3͒ (b) ͑4, 3␲͒ ■ ■ ␲͞3 ഛ ␪ ഛ 2␲͞3 8. r ജ 0, Plot the point whose polar coordinates are given. Then find the Cartesian coordinates of the point. (b) (2 s2, 3␲͞4) ■ 7. 1 ഛ r ഛ 2 ■ |||| 3. (a) ͑3, ␲͞2͒ ■ 7–12 |||| Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. (c) ͑3, 2͒ (b) ͑2, Ϫ␲͞ 7͒ 2. (a) ͑3, 0͒ ■ c=_1 The remaining parts of Figure 18 show that as c becomes negative, the shapes change in reverse order. In fact, these curves are reflections about the horizontal axis of the corresponding curves with positive c. Members of the family of limaçons r=1+c sin ¨ |||| 11.3 c=_ 0.5 c=_ 0.2 5␲͞3 ഛ ␪ ഛ 7␲͞3 ␲͞4 ഛ ␪ ഛ 3␲͞4 12. Ϫ1 ഛ r ഛ 1, ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 13. Find the distance between the points with polar coordinates ͑1, ␲͞6͒ and ͑3, 3␲͞4͒ . 14. Find a formula for the distance between the points with polar coordinates ͑r 1, ␪ 1 ͒ and ͑r 2 , ␪ 2 ͒. ■ 5E-11(pp 706-715) 714 ❙❙❙❙ 15–20 1/18/06 9:52 AM Page 714 CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES |||| Identify the curve by finding a Cartesian equation for the curve. 49. Show that the polar curve r ෇ 4 ϩ 2 sec ␪ (called a conchoid) has the line x ෇ 2 as a vertical asymptote by showing that lim r l Ϯϱ x ෇ 2. Use this fact to help sketch the conchoid. 15. r ෇ 2 16. r cos ␪ ෇ 1 17. r ෇ 3 sin ␪ 18. r ෇ 2 sin ␪ ϩ 2 cos ␪ 19. r ෇ csc ␪ 20. r ෇ tan ␪ sec ␪ ■ ■ 21–26 ■ ■ ■ ■ ■ ■ ■ ■ ■ 50. Show that the curve r ෇ 2 Ϫ csc ␪ (also a conchoid) has the line y ෇ Ϫ1 as a horizontal asymptote by showing that lim r l Ϯϱ y ෇ Ϫ1. Use this fact to help sketch the conchoid. ■ 51. Show that the curve r ෇ sin ␪ tan ␪ (called a cissoid of Diocles) has the line x ෇ 1 as a vertical asymptote. Show also that the curve lies entirely within the vertical strip 0 ഛ x Ͻ 1. Use these facts to help sketch the cissoid. Find a polar equation for the curve represented by the given Cartesian equation. |||| 21. x ෇ 3 22. x 2 ϩ y 2 ෇ 9 23. x ෇ Ϫy 25. x ϩ y ෇ 2cx 2 ■ ■ ■ 53. (a) In Example 11 the graphs suggest that the limaçon Խ Խ r ෇ 1 ϩ c sin ␪ has an inner loop when c Ͼ 1. Prove that this is true, and find the values of ␪ that correspond to the inner loop. (b) From Figure 18 it appears that the limaçon loses its dimple when c ෇ 1 . Prove this. 2 26. x 2 Ϫ y 2 ෇ 1 2 ■ 52. Sketch the curve ͑x 2 ϩ y 2 ͒3 ෇ 4x 2 y 2. 24. x ϩ y ෇ 9 2 ■ ■ ■ ■ ■ ■ ■ ■ 27–28 |||| For each of the described curves, decide if the curve would be more easily given by a polar equation or a Cartesian equation. Then write an equation for the curve. 54. Match the polar equations with the graphs labeled I–VI. Give 27. (a) A line through the origin that makes an angle of ␲͞6 with reasons for your choices. (Don’t use a graphing device.) (a) r ෇ sin͑␪͞2͒ (b) r ෇ sin͑␪͞4͒ (c) r ෇ sec͑3␪ ͒ (d) r ෇ ␪ sin ␪ (e) r ෇ 1 ϩ 4 cos 5␪ (f) r ෇ 1͞s␪ the positive x-axis (b) A vertical line through the point ͑3, 3͒ 28. (a) A circle with radius 5 and center ͑2, 3͒ (b) A circle centered at the origin with radius 4 ■ ■ 29–46 ■ |||| ■ ■ ■ ■ ■ I ■ ■ II III IV ■ V VI ■ Sketch the curve with the given polar equation. 29. ␪ ෇ Ϫ␲͞6 30. r 2 Ϫ 3r ϩ 2 ෇ 0 31. r ෇ sin ␪ 32. r ෇ Ϫ3 cos ␪ 33. r ෇ 2͑1 Ϫ sin ␪ ͒ 34. r ෇ 1 Ϫ 3 cos ␪ ␪ജ0 35. r ෇ ␪, 36. r ෇ ln ␪, ␪ജ1 37. r ෇ sin 2␪ 38. r ෇ 2 cos 3␪ 39. r ෇ 2 cos 4␪ 40. r ෇ sin 5␪ 41. r ෇ 4 cos 2␪ 42. r 2 ෇ sin 2␪ 43. r ෇ 2 cos͑3␪͞2͒ 44. r 2␪ ෇ 1 2 45. r ෇ 1 ϩ 2 cos 2␪ ■ ■ ■ ■ 55–60 |||| Find the slope of the tangent line to the given polar curve at the point specified by the value of ␪. 46. r ෇ 1 ϩ 2 cos͑␪͞2͒ ■ ■ ■ ■ ■ ■ ■ ■ 57. r ෇ 1͞␪, The figure shows the graph of r as a function of ␪ in Cartesian coordinates. Use it to sketch the corresponding polar curve. 47–48 47. |||| 48. r 2 ■ r ■ 56. r ෇ 2 Ϫ sin ␪, ␪෇␲ ■ ■ 58. r ෇ ln ␪, ␪ ෇ ␲͞6 ■ ␪෇e 60. r ෇ sin 3␪, ■ ■ ■ ␪ ෇ ␲͞3 ■ ␪ ෇ ␲͞6 ■ 61. r ෇ 3 cos ␪ π ■ ■ ■ ■ ■ ■ ■ ■ 64. r ෇ e ␪ 65. r ෇ cos 2␪ ■ 62. r ෇ cos ␪ ϩ sin ␪ 63. r ෇ 1 ϩ cos ␪ 2π ¨ π _2 ■ ■ |||| Find the points on the given curve where the tangent line is horizontal or vertical. 0 2π ¨ ■ 61–66 2 0 ■ ␪ ෇ ␲͞6 59. r ෇ 1 ϩ cos ␪, 1 ■ 55. r ෇ 2 sin ␪, 66. r 2 ෇ sin 2␪ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 5E-11(pp 706-715) 1/18/06 9:52 AM Page 715 SECTION 11.4 AREAS AND LENGTHS IN POLAR COORDINATES 0, represents a circle, and find its center and radius. 68. Show that the curves r ෇ a sin ␪ and r ෇ a cos ␪ intersect at ; 80. The astronomer Giovanni Cassini (1625–1712) studied the right angles. 69–74 family of curves with polar equations Use a graphing device to graph the polar curve. Choose ; the parameter interval to make sure that you produce the entire curve. |||| 69. r ෇ 1 ϩ 2 sin͑␪͞2͒ 70. r ෇ s1 Ϫ 0.8 sin ␪ 71. r ෇ e r 4 Ϫ 2c 2 r 2 cos 2␪ ϩ c 4 Ϫ a 4 ෇ 0 where a and c are positive real numbers. These curves are called the ovals of Cassini even though they are oval shaped only for certain values of a and c. (Cassini thought that these curves might represent planetary orbits better than Kepler’s ellipses.) Investigate the variety of shapes that these curves may have. In particular, how are a and c related to each other when the curve splits into two parts? (nephroid of Freeth) 2 sin ␪ (hippopede) Ϫ 2 cos͑4␪ ͒ (butterfly curve) 72. r ෇ sin ͑4␪ ͒ ϩ cos͑4␪ ͒ 2 73. r ෇ 2 Ϫ 5 sin͑␪͞6͒ 81. Let P be any point (except the origin) on the curve r ෇ f ͑␪ ͒. If ␺ is the angle between the tangent line at P and the radial line OP, show that 74. r ෇ cos͑ ␪͞2͒ ϩ cos͑ ␪͞3͒ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ tan ␺ ෇ ; 75. How are the graphs of r ෇ 1 ϩ sin͑␪ Ϫ ␲͞6͒ and r ෇ 1 ϩ sin͑␪ Ϫ ␲͞3͒ related to the graph of r ෇ 1 ϩ sin ␪ ? In general, how is the graph of r ෇ f ͑␪ Ϫ ␣͒ related to the graph of r ෇ f ͑␪ ͒? r dr͞d␪ [Hint: Observe that ␺ ෇ ␾ Ϫ ␪ in the figure.] ; 76. Use a graph to estimate the y-coordinate of the highest points r=f(¨ ) on the curve r ෇ sin 2␪. Then use calculus to find the exact value. ÿ ; 77. (a) Investigate the family of curves defined by the polar equa- P tions r ෇ sin n␪, where n is a positive integer. How is the number of loops related to n ? (b) What happens if the equation in part (a) is replaced by r ෇ sin n␪ ? Խ ¨ Խ where c is a real number and n is a positive integer. How does the graph change as n increases? How does it change as c changes? Illustrate by graphing enough members of the family to support your conclusions. 82. (a) Use Exercise 81 to show that the angle between the tangent ; ; 79. A family of curves has polar equations r෇ ˙ O ; 78. A family of curves is given by the equations r ෇ 1 ϩ c sin n␪, |||| 11.4 715 Investigate how the graph changes as the number a changes. In particular, you should identify the transitional values of a for which the basic shape of the curve changes. 67. Show that the polar equation r ෇ a sin ␪ ϩ b cos ␪, where ab ❙❙❙❙ 1 Ϫ a cos ␪ 1 ϩ a cos ␪ line and the radial line is ␺ ෇ ␲͞4 at every point on the curve r ෇ e ␪. (b) Illustrate part (a) by graphing the curve and the tangent lines at the points where ␪ ෇ 0 and ␲͞2. (c) Prove that any polar curve r ෇ f ͑␪ ͒ with the property that the angle ␺ between the radial line and the tangent line is a constant must be of the form r ෇ Ce k␪, where C and k are constants. Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the area of a sector of a circle r ¨ FIGURE 1 1 A ෇ 1 r 2␪ 2 where, as in Figure 1, r is the radius and ␪ is the radian measure of the central angle. Formula 1 follows from the fact that the area of a sector is proportional to its central angle: A ෇ ͑␪͞2␲͒␲ r 2 ෇ 1 r 2␪. (See also Exercise 35 in Section 8.3.) 2 5E-11(pp 716-725) 716 ❙❙❙❙ 1/18/06 5:30 PM Page 716 CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES Let ᏾ be the region, illustrated in Figure 2, bounded by the polar curve r ෇ f ͑␪ ͒ and by the rays ␪ ෇ a and ␪ ෇ b, where f is a positive continuous function and where 0 Ͻ b Ϫ a ഛ 2␲. We divide the interval ͓a, b͔ into subintervals with endpoints ␪ 0 , ␪1 , ␪ 2 , . . . , ␪ n and equal width ⌬␪. The rays ␪ ෇ ␪ i then divide ᏾ into n smaller regions with central angle ⌬␪ ෇ ␪ i Ϫ ␪ iϪ1 . If we choose ␪ i* in the i th subinterval ͓␪ iϪ1, ␪ i ͔, then the area ⌬Ai of the i th region is approximated by the area of the sector of a circle with central angle ⌬␪ and radius f ͑␪ i*͒. (See Figure 3.) Thus, from Formula 1 we have r=f(¨) ᏾ ¨=b b O ¨=a a FIGURE 2 f(¨ i*) ⌬Ai Ϸ 1 ͓ f ͑␪ i*͔͒ 2 ⌬␪ 2 ¨=¨ i ¨=¨ i-1 and so an approximation to the total area A of ᏾ is n ¨=b AϷ 2 ͚ 1 2 ͓ f ͑␪ i*͔͒ 2 ⌬␪ i෇1 Ψ ¨=a It appears from Figure 3 that the approximation in (2) improves as n l ϱ. But the sums in (2) are Riemann sums for the function t͑␪ ͒ ෇ 1 ͓ f ͑␪ ͔͒ 2, so 2 O FIGURE 3 n lim ͚ n l ϱ i෇1 1 2 b 1 2 a ͓ f ͑␪ i*͔͒ 2 ⌬␪ ෇ y ͓ f ͑␪ ͔͒ 2 d␪ It therefore appears plausible (and can in fact be proved) that the formula for the area A of the polar region ᏾ is A෇y 3 b 1 2 a ͓ f ͑␪ ͔͒ 2 d␪ Formula 3 is often written as A෇y 4 b 1 2 a r 2 d␪ with the understanding that r ෇ f ͑␪ ͒. Note the similarity between Formulas 1 and 4. When we apply Formula 3 or 4 it is helpful to think of the area as being swept out by a rotating ray through O that starts with angle a and ends with angle b. EXAMPLE 1 Find the area enclosed by one loop of the four-leaved rose r ෇ cos 2␪. r=cos 2¨ π ¨= 4 SOLUTION The curve r ෇ cos 2␪ was sketched in Example 8 in Section 11.3. Notice from Figure 4 that the region enclosed by the right loop is swept out by a ray that rotates from ␪ ෇ Ϫ␲͞4 to ␪ ෇ ␲͞4. Therefore, Formula 4 gives A෇y ␲͞4 1 2 Ϫ␲͞4 π ¨=_ 4 FIGURE 4 ෇y ␲͞4 0 [ r 2 d␪ ෇ 1 y 2 ␲͞4 Ϫ␲͞4 cos 2 2␪ d␪ ෇ y ෇ 1 ␪ ϩ 1 sin 4␪ 2 4 cos 2 2␪ d␪ ␲͞4 1 2 0 ␲͞4 0 ෇ ␲ 8 ͑1 ϩ cos 4␪ ͒ d␪ 5E-11(pp 716-725) 1/18/06 5:31 PM Page 717 SECTION 11.4 AREAS AND LENGTHS IN POLAR COORDINATES ❙❙❙❙ 717 EXAMPLE 2 Find the area of the region that lies inside the circle r ෇ 3 sin ␪ and outside the cardioid r ෇ 1 ϩ sin ␪. SOLUTION The cardioid (see Example 7 in Section 11.3) and the circle are sketched in Figure 5 and the desired region is shaded. The values of a and b in Formula 4 are determined by finding the points of intersection of the two curves. They intersect when 3 sin ␪ ෇ 1 ϩ sin ␪, which gives sin ␪ ෇ 1 , so ␪ ෇ ␲͞6, 5␲͞6. The desired area can be 2 found by subtracting the area inside the cardioid between ␪ ෇ ␲͞6 and ␪ ෇ 5␲͞6 from the area inside the circle from ␲͞6 to 5␲͞6. Thus r=3 sin ¨ π 5π ¨= 6 ¨= 6 O A෇1y 2 r=1+sin ¨ 5␲͞6 ␲͞6 FIGURE 5 ͑3 sin ␪ ͒2 d␪ Ϫ 1 y 2 5␲͞6 ␲͞6 ͑1 ϩ sin ␪ ͒2 d␪ Since the region is symmetric about the vertical axis ␪ ෇ ␲͞2, we can write ͫy A෇2 ෇y 1 2 ␲ ͞2 ␲ ͞6 ෇y ␲ ͞2 ␲ ͞6 ␲ ͞2 ␲ ͞6 9 sin 2␪ d␪ Ϫ 1 y 2 ͬ ͑3 Ϫ 4 cos 2␪ Ϫ 2 sin ␪ ͒ d␪ ␲ ͞2 ␲ ͞6 [because sin 2␪ ෇ 1 ͑1 Ϫ cos 2␪ ͒] 2 ෇␲ Example 2 illustrates the procedure for finding the area of the region bounded by two polar curves. In general, let ᏾ be a region, as illustrated in Figure 6, that is bounded by curves with polar equations r ෇ f ͑␪ ͒, r ෇ t͑␪ ͒, ␪ ෇ a, and ␪ ෇ b, where f ͑␪ ͒ ജ t͑␪ ͒ ജ 0 and 0 Ͻ b Ϫ a ഛ 2␲. The area A of ᏾ is found by subtracting the area inside r ෇ t͑␪ ͒ from the area inside r ෇ f ͑␪ ͒, so using Formula 3 we have ᏾ ¨=b ¨=a ͑1 ϩ 2 sin ␪ ϩ sin 2␪ ͒ d␪ ͑8 sin 2␪ Ϫ 1 Ϫ 2 sin ␪ ͒ d␪ ෇ 3␪ Ϫ 2 sin 2␪ ϩ 2 cos ␪ r=f(¨) ␲ ͞2 ␲ ͞6 r=g(¨) O A෇y b 1 2 a ͓ f ͑␪ ͔͒ 2 d␪ Ϫ y b 1 2 a ͓t͑␪ ͔͒ 2 d␪ FIGURE 6 b ෇ 1 y ͓͑ f ͑␪ ͔͒ 2 Ϫ ͓t͑␪ ͔͒ 2 ͒ d␪ 2 a | CAUTION The fact that a single point has many representations in polar coordinates sometimes makes it difficult to find all the points of intersection of two polar curves. For instance, it is obvious from Figure 5 that the circle and the cardioid have three points of intersection; however, in Example 2 we solved the equations r ෇ 3 sin ␪ and r ෇ 1 ϩ sin ␪ and found only two such points, ( 3, ␲͞6) and ( 3, 5␲͞6). The origin is also 2 2 a point of intersection, but we can’t find it by solving the equations of the curves because the origin has no single representation in polar coordinates that satisfies both equations. Notice that, when represented as ͑0, 0͒ or ͑0, ␲͒, the origin satisfies r ෇ 3 sin ␪ and so it lies on the circle; when represented as ͑0, 3␲͞2͒, it satisfies r ෇ 1 ϩ sin ␪ and so it lies on the cardioid. Think of two points moving along the curves as the parameter value ␪ increases from 0 to 2␲. On one curve the origin is reached at ␪ ෇ 0 and ␪ ෇ ␲ ; on the other curve it is reached at ␪ ෇ 3␲͞2. The points don’t collide at the origin because they reach the origin at different times, but the curves intersect there nonetheless. Thus, to find all points of intersection of two polar curves, it is recommended that you draw the graphs of both curves. It is especially convenient to use a graphing calculator or computer to help with this task. ■ 5E-11(pp 716-725) 718 ❙❙❙❙ 1/18/06 5:32 PM Page 718 CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES EXAMPLE 3 Find all points of intersection of the curves r ෇ cos 2␪ and r ෇ 2 . 1 1 π ” , ’ 2 3 1 r=2 1 π ” 2 ’ , 6 r=cos 2¨ SOLUTION If we solve the equations r ෇ cos 2␪ and r ෇ 2 , we get cos 2␪ ෇ 1 1 2 and, therefore, 2␪ ෇ ␲͞3, 5␲͞3, 7␲͞3, 11␲͞3. Thus, the values of ␪ between 0 and 2␲ that satisfy both equations are ␪ ෇ ␲͞6, 5␲͞6, 7␲͞6, 11␲͞6. We have found four points of intersection: ( 1, ␲͞6), ( 1, 5␲͞6), ( 1, 7␲͞6), and ( 1, 11␲͞6). 2 2 2 2 However, you can see from Figure 7 that the curves have four other points of intersection—namely, ( 1, ␲͞3), ( 1, 2␲͞3), ( 1, 4␲͞3), and ( 1, 5␲͞3). These can be found using 2 2 2 2 symmetry or by noticing that another equation of the circle is r ෇ Ϫ 1 and then solving 2 the equations r ෇ cos 2␪ and r ෇ Ϫ 1 . 2 Arc Length FIGURE 7 To find the length of a polar curve r ෇ f ͑␪ ͒, a ഛ ␪ ഛ b, we regard ␪ as a parameter and write the parametric equations of the curve as x ෇ r cos ␪ ෇ f ͑␪ ͒ cos ␪ y ෇ r sin ␪ ෇ f ͑␪ ͒ sin ␪ Using the Product Rule and differentiating with respect to ␪, we obtain dx dr ෇ cos ␪ Ϫ r sin ␪ d␪ d␪ dy dr ෇ sin ␪ ϩ r cos ␪ d␪ d␪ so, using cos 2␪ ϩ sin 2␪ ෇ 1, we have ͩ ͪ ͩ ͪ ͩ ͪ dx d␪ 2 dy d␪ ϩ 2 2 dr d␪ ෇ ෇ ϩ ͩ ͪ cos 2␪ Ϫ 2r ͩ ͪ dr d␪ 2 dr cos ␪ sin ␪ ϩ r 2 sin 2␪ d␪ sin 2␪ ϩ 2r dr sin ␪ cos ␪ ϩ r 2 cos 2␪ d␪ dr 2 ϩ r2 d␪ Assuming that f Ј is continuous, we can use Theorem 11.2.6 to write the arc length as ෇ L෇ y ͱͩ ͪ ͩ ͪ dx d␪ b a 2 2 dy d␪ ϩ d␪ Therefore, the length of a curve with polar equation r ෇ f ͑␪ ͒, a ഛ ␪ ഛ b, is L෇ 5 y b a ͱ ͩ ͪ r2 ϩ dr d␪ 2 d␪ EXAMPLE 4 Find the length of the cardioid r ෇ 1 ϩ sin ␪. SOLUTION The cardioid is shown in Figure 8. (We sketched it in Example 7 in Section 11.3.) Its full length is given by the parameter interval 0 ഛ ␪ ഛ 2␲, so Formula 5 gives O L෇ y 2␲ 0 FIGURE 8 r=1+sin ¨ ෇y 2␲ 0 ͱ ͩ ͪ r2 ϩ dr d␪ 2 d␪ ෇ y s2 ϩ 2 sin ␪ d␪ 2␲ 0 s͑1 ϩ sin ␪ ͒2 ϩ cos 2␪ d␪ 5E-11(pp 716-725) 1/18/06 5:33 PM Page 719 SECTION 11.4 AREAS AND LENGTHS IN POLAR COORDINATES ❙❙❙❙ 719 We could evaluate this integral by multiplying and dividing the integrand by s2 Ϫ 2 sin ␪ , or we could use a computer algebra system. In any event, we find that the length of the cardioid is L ෇ 8. |||| 11.4 Exercises 1–4 |||| Find the area of the region that is bounded by the given curve and lies in the specified sector. 1. r ෇ s␪ , ■ 0 ഛ ␪ ഛ ␲͞4 2. r ෇ e ␪͞2, 21. r ෇ 1 ϩ 2 sin ␪ ␲ ഛ ␪ ഛ 2␲ 3. r ෇ sin ␪, ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 23. r ෇ 4 sin ␪, ■ ■ ■ ■ ■ ■ 5. 27. r ෇ 3 cos ␪, 6. ■ ■ 29–34 ■ 34. r ෇ a sin ␪, ■ ■ ■ ■ ■ ■ 30. r ෇ sin 2␪, r ෇ cos 2␪ 33. r ෇ 3 ϩ 2 sin ␪, 8. 7. ■ r ෇ cos ␪ 31. r ෇ sin 2␪, r=1+sin ¨ r ෇ 3 cos ␪ ■ ■ ■ Find the area of the region that lies inside both curves. 29. r ෇ sin ␪ , r=¨ r ෇ 3 sin ␪ r ෇ 1 ϩ cos ␪ ■ |||| r෇1 26. r ෇ 2 ϩ sin ␪, r෇2 28. r ෇ 1 ϩ cos ␪, ■ 24. r ෇ 1 Ϫ sin ␪, r෇2 25. r ෇ 8 cos 2␪, Find the area of the shaded region. |||| ■ 23–28 |||| Find the area of the region that lies inside the first curve and outside the second curve. 2 5–8 (inner loop) ■ r ෇ 2 cos ␪ Ϫ sec ␪. 0ഛ␪ഛ␲ ■ ■ 22. Find the area enclosed by the loop of the strophoid ␲͞3 ഛ ␪ ഛ 2␲͞3 4. r ෇ ssin ␪ , ■ ■ r ෇ sin ␪ 32. r ෇ 2 sin 2␪, 2 r෇1 r෇2 r ෇ b cos ␪, a Ͼ 0, b Ͼ 0 ■ ■ ■ ■ ■ ■ ■ ■ ■ 35. Find the area inside the larger loop and outside the smaller loop 1 of the limaçon r ෇ 2 ϩ cos ␪. 36. Find the area between a large loop and the enclosed small loop of the curve r ෇ 1 ϩ 2 cos 3␪. r=sin 4¨ r=4+3 sin ¨ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 37–42 ■ |||| Find all points of intersection of the given curves. ■ 37. r ෇ sin ␪, 9–14 |||| Sketch the curve and find the area that it encloses. 9. r ෇ 3 cos ␪ ■ ; 15–16 ■ |||| ■ 40. r ෇ cos 3␪, 41. r ෇ sin ␪, 14. r ෇ 2 ϩ cos 2␪ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 17–21 ■ ■ |||| Find the area of the region enclosed by one loop of the curve. 17. r ෇ sin 2␪ 18. r ෇ 4 sin 3␪ 19. r ෇ 3 cos 5␪ 20. r ෇ 2 cos 4␪ r ෇ sin 3␪ r ෇ sin 2␪ ■ ■ r 2 ෇ cos 2␪ ■ ■ ■ ■ ■ ■ ■ ■ ■ ; 43. The points of intersection of the cardioid r ෇ 1 ϩ sin ␪ and the 16. r ෇ 2 sin ␪ ϩ 3 sin 9␪ ■ r ෇ 1 Ϫ cos ␪ 42. r ෇ sin 2␪, 2 ■ Graph the curve and find the area that it encloses. 15. r ෇ 1 ϩ 2 sin 6␪ ■ 39. r ෇ cos ␪, 12. r 2 ෇ sin 2␪ 13. r ෇ 2 cos 3␪ ■ r ෇ 2 cos 2␪ 38. r ෇ 2, 10. r ෇ 3͑1 ϩ cos ␪ ͒ 11. r 2 ෇ 4 cos 2␪ r ෇ cos ␪ ■ spiral loop r ෇ 2␪, Ϫ␲͞2 ഛ ␪ ഛ ␲͞2, can’t be found exactly. Use a graphing device to find the approximate values of ␪ at which they intersect. Then use these values to estimate the area that lies inside both curves. ; 44. Use a graph to estimate the values of ␪ for which the curves r ෇ 3 ϩ sin 5␪ and r ෇ 6 sin ␪ intersect. Then estimate the area that lies inside both curves. 5E-11(pp 716-725) 720 ❙❙❙❙ 45–48 1/18/06 |||| 55. (a) Use Formula 11.2.7 to show that the area of the surface Find the exact length of the polar curve. 47. r ෇ ␪ 2, ■ 49–52 Page 720 CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES 45. r ෇ 3 sin ␪, ■ 5:33 PM 0 ഛ ␪ ഛ ␲͞3 0 ഛ ␪ ഛ 2␲ ■ ■ ■ 2␪ 48. r ෇ ␪, ■ ■ ■ generated by rotating the polar curve 0 ഛ ␪ ഛ 2␲ 46. r ෇ e , r ෇ f ͑␪ ͒ 0 ഛ ␪ ഛ 2␲ ■ ■ ■ ■ (where f Ј is continuous and 0 ഛ a Ͻ b ഛ ␲) about the polar axis is Use a calculator to find the length of the curve correct to four decimal places. |||| 49. r ෇ 3 sin 2␪ 51. r ෇ sin͑␪͞2͒ ■ ■ b S ෇ y 2␲ r sin ␪ a 50. r ෇ 4 sin 3␪ ■ ■ ■ ■ ■ ■ ■ ͱ ͩ ͪ r2 ϩ dr d␪ 2 d␪ (b) Use the formula in part (a) to find the surface area generated by rotating the lemniscate r 2 ෇ cos 2␪ about the polar axis. 52. r ෇ 1 ϩ cos͑␪͞3͒ ■ aഛ␪ഛb ■ ■ 56. (a) Find a formula for the area of the surface generated by ; 53–54 |||| Graph the curve and find its length. 53. r ෇ cos 4͑␪͞4͒ ■ ■ ■ |||| 11.5 54. r ෇ cos 2͑␪͞2͒ ■ ■ ■ ■ ■ ■ ■ ■ ■ rotating the polar curve r ෇ f ͑␪ ͒, a ഛ ␪ ഛ b (where f Ј is continuous and 0 ഛ a Ͻ b ഛ ␲), about the line ␪ ෇ ␲͞2. (b) Find the surface area generated by rotating the lemniscate r 2 ෇ cos 2␪ about the line ␪ ෇ ␲͞2. Conic Sections In this section we give geometric definitions of parabolas, ellipses, and hyperbolas and derive their standard equations. They are called conic sections, or conics, because they result from intersecting a cone with a plane as shown in Figure 1. ellipse parabola hyperbola FIGURE 1 Conics parabola axis focus vertex FIGURE 2 F directrix Parabolas A parabola is the set of points in a plane that are equidistant from a fixed point F (called the focus) and a fixed line (called the directrix). This definition is illustrated by Figure 2. Notice that the point halfway between the focus and the directrix lies on the parabola; it is called the vertex. The line through the focus perpendicular to the directrix is called the axis of the parabola. In the 16th century Galileo showed that the path of a projectile that is shot into the air at an angle to the ground is a parabola. Since then, parabolic shapes have been used in designing automobile headlights, reflecting telescopes, and suspension bridges. 5E-11(pp 716-725) 1/18/06 5:34 PM Page 721 SECTION 11.5 CONIC SECTIONS y P(x, y) F(0, p) y p x O y=_p ❙❙❙❙ 721 (See Problem 14 on page 220 for the reflection property of parabolas that makes them so useful.) We obtain a particularly simple equation for a parabola if we place its vertex at the origin O and its directrix parallel to the x-axis as in Figure 3. If the focus is the point ͑0, p͒, then the directrix has the equation y ෇ Ϫp. If P͑x, y͒ is any point on the parabola, then the distance from P to the focus is Խ PF Խ ෇ sx ϩ ͑y Ϫ p͒ and the distance from P to the directrix is Խ y ϩ p Խ. (Figure 3 illustrates the case where 2 2 p Ͼ 0.) The defining property of a parabola is that these distances are equal: Խ FIGURE 3 sx 2 ϩ ͑y Ϫ p͒2 ෇ y ϩ p Խ We get an equivalent equation by squaring and simplifying: Խ x 2 ϩ ͑y Ϫ p͒2 ෇ y ϩ p Խ 2 ෇ ͑y ϩ p͒2 x 2 ϩ y 2 Ϫ 2py ϩ p 2 ෇ y 2 ϩ 2py ϩ p 2 x 2 ෇ 4py An equation of the parabola with focus ͑0, p͒ and directrix y ෇ Ϫp is 1 x 2 ෇ 4py If we write a ෇ 1͑͞4p͒, then the standard equation of a parabola (1) becomes y ෇ ax 2. It opens upward if p Ͼ 0 and downward if p Ͻ 0 [see Figure 4, parts (a) and (b)]. The graph is symmetric with respect to the y-axis because (1) is unchanged when x is replaced by Ϫx. y y y y=_p (0, p) x x (0, p) y=_p (a) ≈=4py, p>0 ( p, 0) ( p, 0) 0 0 y (b) ≈=4py, p<0 0 x x 0 x=_p x=_p (c) ¥=4px, p>0 (d) ¥=4px, p<0 FIGURE 4 If we interchange x and y in (1), we obtain 2 y 2 ෇ 4px which is an equation of the parabola with focus ͑p, 0͒ and directrix x ෇ Ϫp. (Interchanging x and y amounts to reflecting about the diagonal line y ෇ x.) The parabola opens to the right if p Ͼ 0 and to the left if p Ͻ 0 [see Figure 4, parts (c) and (d)]. In both cases the graph is symmetric with respect to the x-axis, which is the axis of the parabola. 5E-11(pp 716-725) 722 ❙❙❙❙ 1/18/06 5:34 PM Page 722 CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES y EXAMPLE 1 Find the focus and directrix of the parabola y 2 ϩ 10x ෇ 0 and sketch ¥+10x=0 the graph. SOLUTION If we write the equation as y 2 ෇ Ϫ10x and compare it with Equation 2, we see that 4p ෇ Ϫ10, so p ෇ Ϫ5 . Thus, the focus is ͑ p, 0͒ ෇ (Ϫ 5, 0) and the directrix is 2 2 x ෇ 5 . The sketch is shown in Figure 5. 2 ”_ 5 , 0’ 2 x 0 Ellipses 5 x= 2 An ellipse is the set of points in a plane the sum of whose distances from two fixed points F1 and F2 is a constant (see Figure 6). These two fixed points are called the foci (plural of focus). One of Kepler’s laws is that the orbits of the planets in the solar system are ellipses with the Sun at one focus. In order to obtain the simplest equation for an ellipse, we place the foci on the x-axis at the points ͑Ϫc, 0͒ and ͑c, 0͒ as in Figure 7 so that the origin is halfway between the foci. Let the sum of the distances from a point on the ellipse to the foci be 2a Ͼ 0. Then P͑x, y͒ is a point on the ellipse when PF1 ϩ PF2 ෇ 2a FIGURE 5 P F¡ F™ Խ FIGURE 6 Խ Խ Խ that is, or y P(x, y) s͑x ϩ c͒2 ϩ y 2 ϩ s͑x Ϫ c͒2 ϩ y 2 ෇ 2a s͑x Ϫ c͒2 ϩ y 2 ෇ 2a Ϫ s͑x ϩ c͒2 ϩ y 2 Squaring both sides, we have F¡(_c, 0) 0 F™(c, 0) x 2 Ϫ 2cx ϩ c 2 ϩ y 2 ෇ 4a 2 Ϫ 4as͑x ϩ c͒2 ϩ y 2 ϩ x 2 ϩ 2cx ϩ c 2 ϩ y 2 x which simplifies to as͑x ϩ c͒2 ϩ y 2 ෇ a 2 ϩ cx We square again: FIGURE 7 a 2͑x 2 ϩ 2cx ϩ c 2 ϩ y 2 ͒ ෇ a 4 ϩ 2a 2cx ϩ c 2x 2 which becomes ͑a 2 Ϫ c 2 ͒x 2 ϩ a 2 y 2 ෇ a 2͑a 2 Ϫ c 2 ͒ From triangle F1 F2 P in Figure 7 we see that 2c Ͻ 2a, so c Ͻ a and, therefore, a 2 Ϫ c 2 Ͼ 0. For convenience, let b 2 ෇ a 2 Ϫ c 2. Then the equation of the ellipse becomes b 2x 2 ϩ a 2 y 2 ෇ a 2b 2 or, if both sides are divided by a 2b 2, x2 y2 ϩ 2 ෇1 a2 b 3 y (0, b) (_a, 0) a b (_c, 0) 0 c (0, _b) FIGURE 8 ≈ ¥ + =1 a@ b@ (a, 0) (c, 0) x Since b 2 ෇ a 2 Ϫ c 2 Ͻ a 2, it follows that b Ͻ a. The x-intercepts are found by setting y ෇ 0. Then x 2͞a 2 ෇ 1, or x 2 ෇ a 2, so x ෇ Ϯa. The corresponding points ͑a, 0͒ and ͑Ϫa, 0͒ are called the vertices of the ellipse and the line segment joining the vertices is called the major axis. To find the y-intercepts we set x ෇ 0 and obtain y 2 ෇ b 2, so y ෇ Ϯb. Equation 3 is unchanged if x is replaced by Ϫx or y is replaced by Ϫy, so the ellipse is symmetric about both axes. Notice that if the foci coincide, then c ෇ 0, so a ෇ b and the ellipse becomes a circle with radius r ෇ a ෇ b. We summarize this discussion as follows (see also Figure 8). 4 The ellipse y2 x2 ϩ 2 ෇1 a2 b aജbϾ0 has foci ͑Ϯc, 0͒, where c 2 ෇ a 2 Ϫ b 2, and vertices ͑Ϯa, 0͒. 5E-11(pp 716-725) 1/18/06 5:35 PM Page 723 SECTION 11.5 CONIC SECTIONS y ❙❙❙❙ 723 If the foci of an ellipse are located on the y-axis at ͑0, Ϯc͒, then we can find its equation by interchanging x and y in (4). (See Figure 9.) (0, a) (0, c) 5 (_b, 0) The ellipse (b, 0) 0 x2 y2 ෇1 2 ϩ b a2 x (0, _c) aജbϾ0 has foci ͑0, Ϯc͒, where c 2 ෇ a 2 Ϫ b 2, and vertices ͑0, Ϯa͒. (0, _a) EXAMPLE 2 Sketch the graph of 9x 2 ϩ 16y 2 ෇ 144 and locate the foci. FIGURE 9 ≈ ¥ + =1, a˘b b@ a@ SOLUTION Divide both sides of the equation by 144: x2 y2 ϩ ෇1 16 9 The equation is now in the standard form for an ellipse, so we have a 2 ෇ 16, b 2 ෇ 9, a ෇ 4, and b ෇ 3. The x-intercepts are Ϯ4 and the y-intercepts are Ϯ3. Also, c 2 ෇ a 2 Ϫ b 2 ෇ 7, so c ෇ s7 and the foci are (Ϯs7, 0). The graph is sketched in Figure 10. y (0, 3) (_4, 0) {_œ„, 0} 7 (4, 0) 0 {œ„, 0} 7 x EXAMPLE 3 Find an equation of the ellipse with foci ͑0, Ϯ2͒ and vertices ͑0, Ϯ3͒. SOLUTION Using the notation of (5), we have c ෇ 2 and a ෇ 3. Then we obtain b 2 ෇ a 2 Ϫ c 2 ෇ 9 Ϫ 4 ෇ 5, so an equation of the ellipse is (0, _3) x2 y2 ϩ ෇1 5 9 FIGURE 10 9≈+16¥=144 Another way of writing the equation is 9x 2 ϩ 5y 2 ෇ 45. Like parabolas, ellipses have an interesting reflection property that has practical consequences. If a source of light or sound is placed at one focus of a surface with elliptical cross-sections, then all the light or sound is reflected off the surface to the other focus (see Exercise 59). This principle is used in lithotripsy, a treatment for kidney stones. A reflector with elliptical cross-section is placed in such a way that the kidney stone is at one focus. High-intensity sound waves generated at the other focus are reflected to the stone and destroy it without damaging surrounding tissue. The patient is spared the trauma of surgery and recovers within a few days. y Hyperbolas P(x, y) F¡(_c, 0) 0 FIGURE 11 P is on the hyperbola when | PF¡|-| PF™ |=Ϯ2a F™(c, 0) x A hyperbola is the set of all points in a plane the difference of whose distances from two fixed points F1 and F2 (the foci) is a constant. This definition is illustrated in Figure 11. Hyperbolas occur frequently as graphs of equations in chemistry, physics, biology, and economics (Boyle’s Law, Ohm’s Law, supply and demand curves). A particularly significant application of hyperbolas is found in the navigation systems developed in World Wars I and II (see Exercise 51). Notice that the definition of a hyperbola is similar to that of an ellipse; the only change is that the sum of distances has become a difference of distances. In fact, the derivation of the equation of a hyperbola is also similar to the one given earlier for an ellipse. It is left 5E-11(pp 716-725) 724 ❙❙❙❙ 1/18/06 5:35 PM Page 724 CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES as Exercise 52 to show that when the foci are on the x-axis at ͑Ϯc, 0͒ and the difference of distances is PF1 Ϫ PF2 ෇ Ϯ2a, then the equation of the hyperbola is Խ Խ Խ Խ x2 y2 ෇1 2 Ϫ a b2 6 b y where c 2 ෇ a 2 ϩ b 2. Notice that the x-intercepts are again Ϯa and the points ͑a, 0͒ and ͑Ϫa, 0͒ are the vertices of the hyperbola. But if we put x ෇ 0 in Equation 6 we get y 2 ෇ Ϫb 2, which is impossible, so there is no y-intercept. The hyperbola is symmetric with respect to both axes. To analyze the hyperbola further, we look at Equation 6 and obtain b y=_ a x y= a x (_a, 0) (a, 0) (_c, 0) (c, 0) 0 x2 y2 ജ1 2 ෇ 1 ϩ a b2 x Խ Խ This shows that x 2 ജ a 2, so x ෇ sx 2 ജ a. Therefore, we have x ജ a or x ഛ Ϫa. This means that the hyperbola consists of two parts, called its branches. When we draw a hyperbola it is useful to first draw its asymptotes, which are the dashed lines y ෇ ͑b͞a͒x and y ෇ Ϫ͑b͞a͒x shown in Figure 12. Both branches of the hyperbola approach the asymptotes; that is, they come arbitrarily close to the asymptotes. [See Exercise 55 in Section 4.5, where y ෇ ͑b͞a͒x is shown to be a slant asymptote.] FIGURE 12 ≈ ¥ - =1 a@ b@ 7 The hyperbola x2 y2 Ϫ 2 ෇1 a2 b y (0, c) a y=_ b x a has foci ͑Ϯc, 0͒, where c 2 ෇ a 2 ϩ b 2, vertices ͑Ϯa, 0͒, and asymptotes y ෇ Ϯ͑b͞a͒x. y= b x (0, a) (0, _a) 0 x If the foci of a hyperbola are on the y-axis, then by reversing the roles of x and y we obtain the following information, which is illustrated in Figure 13. (0, _c) 8 FIGURE 13 ¥ ≈ - =1 a@ b@ The hyperbola y2 x2 ෇1 2 Ϫ a b2 has foci ͑0, Ϯc͒, where c 2 ෇ a 2 ϩ b 2, vertices ͑0, Ϯa͒, and asymptotes y ෇ Ϯ͑a͞b͒x. 3 y=_ 4 x y 3 y= 4 x EXAMPLE 4 Find the foci and asymptotes of the hyperbola 9x 2 Ϫ 16y 2 ෇ 144 and sketch its graph. (_4, 0) (_5, 0) SOLUTION If we divide both sides of the equation by 144, it becomes (4, 0) 0 FIGURE 14 9≈-16¥=144 (5, 0) x x2 y2 Ϫ ෇1 16 9 which is of the form given in (7) with a ෇ 4 and b ෇ 3. Since c 2 ෇ 16 ϩ 9 ෇ 25, the foci are ͑Ϯ5, 0͒. The asymptotes are the lines y ෇ 3 x and y ෇ Ϫ 3 x. The graph is shown 4 4 in Figure 14. 5E-11(pp 716-725) 1/18/06 5:36 PM Page 725 SECTION 11.5 CONIC SECTIONS ❙❙❙❙ 725 EXAMPLE 5 Find the foci and equation of the hyperbola with vertices ͑0, Ϯ1͒ and asymptote y ෇ 2x. SOLUTION From (8) and the given information, we see that a ෇ 1 and a͞b ෇ 2. Thus, 1 5 b ෇ a͞2 ෇ 2 and c 2 ෇ a 2 ϩ b 2 ෇ 4 . The foci are (0, Ϯs5͞2) and the equation of the hyperbola is y 2 Ϫ 4x 2 ෇ 1 Shifted Conics As discussed in Appendix C, we shift conics by taking the standard equations (1), (2), (4), (5), (7), and (8) and replacing x and y by x Ϫ h and y Ϫ k. EXAMPLE 6 Find an equation of the ellipse with foci ͑2, Ϫ2͒, ͑4, Ϫ2͒ and vertices ͑1, Ϫ2͒, ͑5, Ϫ2͒. SOLUTION The major axis is the line segment that joins the vertices ͑1, Ϫ2͒, ͑5, Ϫ2͒ and has length 4, so a ෇ 2. The distance between the foci is 2, so c ෇ 1. Thus, b 2 ෇ a 2 Ϫ c 2 ෇ 3. Since the center of the ellipse is ͑3, Ϫ2͒, we replace x and y in (4) by x Ϫ 3 and y ϩ 2 to obtain ͑x Ϫ 3͒2 ͑y ϩ 2͒2 ϩ ෇1 4 3 as the equation of the ellipse. EXAMPLE 7 Sketch the conic 9x 2 Ϫ 4y 2 Ϫ 72x ϩ 8y ϩ 176 ෇ 0 y and find its foci. 3 y-1=_ 2 (x-4) SOLUTION We complete the squares as follows: 4͑y 2 Ϫ 2y͒ Ϫ 9͑x 2 Ϫ 8x͒ ෇ 176 (4, 4) 4͑y 2 Ϫ 2y ϩ 1͒ Ϫ 9͑x 2 Ϫ 8x ϩ 16͒ ෇ 176 ϩ 4 Ϫ 144 (4, 1) 4͑y Ϫ 1͒2 Ϫ 9͑x Ϫ 4͒2 ෇ 36 x 0 (4, _2) 3 y-1= 2 (x-4) FIGURE 15 9≈-4¥-72x+8y+176=0 ͑y Ϫ 1͒2 ͑x Ϫ 4͒2 Ϫ ෇1 9 4 This is in the form (8) except that x and y are replaced by x Ϫ 4 and y Ϫ 1. Thus, a 2 ෇ 9, b 2 ෇ 4, and c 2 ෇ 13. The hyperbola is shifted four units to the right and one unit upward. The foci are (4, 1 ϩ s13 ) and (4, 1 Ϫ s13 ) and the vertices are ͑4, 4͒ and ͑4, Ϫ2͒. The asymptotes are y Ϫ 1 ෇ Ϯ3 ͑x Ϫ 4͒. The hyperbola is sketched in 2 Figure 15. 5E-11(pp 726-735) ❙❙❙❙ 726 1/18/06 Page 726 CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES |||| 11.5 1–8 9:35 AM Exercises 23. 2y 2 Ϫ 3x 2 Ϫ 4y ϩ 12x ϩ 8 ෇ 0 Find the vertex, focus, and directrix of the parabola and sketch its graph. |||| 1. x ෇ 2y 24. 16x 2 Ϫ 9y 2 ϩ 64x Ϫ 90y ෇ 305 2. 4y ϩ x ෇ 0 2 2 ■ 3. 4x ෇ Ϫy 4. y ෇ 12x 5. ͑x ϩ 2͒ ෇ 8͑ y Ϫ 3͒ 6. x Ϫ 1 ෇ ͑ y ϩ 5͒2 7. y 2 ϩ 2y ϩ 12x ϩ 25 ෇ 0 8. y ϩ 12 x Ϫ 2x 2 ෇ 16 2 2 ■ ■ 9–10 ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 9. ■ ■ ■ ■ 10. y ■ ■ 25. x 2 ෇ y ϩ 1 ■ 26. x 2 ෇ y 2 ϩ 1 27. x 2 ෇ 4y Ϫ 2y 2 28. y 2 Ϫ 8y ෇ 6x Ϫ 16 29. y 2 ϩ 2y ෇ 4x 2 ϩ 3 ■ 30. 4x 2 ϩ 4x ϩ y 2 ෇ 0 ■ y ■ ■ ■ ■ ■ ■ ■ ■ 31. Parabola, 0 ■ ■ ■ ■ x 2 ■ ■ ■ ■ ■ vertex ͑0, 0͒, vertex ͑1, 0͒, directrix x ෇ Ϫ5 focus ͑Ϫ4, 0͒, directrix x ෇ 2 34. Parabola, focus ͑3, 6͒, vertex ͑3, 2͒ vertex ͑0, 0͒, passing through (1,Ϫ4) 35. Parabola, 11–16 |||| ■ focus ͑0, Ϫ2͒ 32. Parabola, 1 x ■ |||| Find an equation for the conic that satisfies the given conditions. 33. Parabola, _2 ■ 31–48 1 ■ ■ |||| Identify the type of conic section whose equation is given and find the vertices and foci. |||| ■ ■ 25–30 Find an equation of the parabola. Then find the focus and directrix. ■ ■ 2 Find the vertices and foci of the ellipse and sketch its graph. 36. Parabola, axis the x-axis, passing through ͑Ϫ2, 3͒, ͑0, 3͒, vertical axis, and ͑1, 9͒ x2 y2 ϩ ෇1 11. 9 5 x2 y2 ϩ ෇1 12. 64 100 37. Ellipse, foci ͑Ϯ2, 0͒, vertices ͑Ϯ5, 0͒ 13. 4x 2 ϩ y 2 ෇ 16 14. 4x 2 ϩ 25y 2 ෇ 25 38. Ellipse, foci ͑0, Ϯ5͒, vertices ͑0, Ϯ13͒ 39. Ellipse, foci ͑0, 2͒, ͑0, 6͒ vertices ͑0, 0͒, ͑0, 8͒ 40. Ellipse, foci ͑0, Ϫ1͒, ͑8, Ϫ1͒, 41. Ellipse, center ͑2, 2͒, focus ͑0, 2͒, 42. Ellipse, foci ͑Ϯ2, 0͒, passing through ͑2, 1͒ 15. 9x 2 Ϫ 18x ϩ 4y 2 ෇ 27 16. x 2 ϩ 2y 2 Ϫ 6x ϩ 4y ϩ 7 ෇ 0 ■ ■ 17–18 ■ |||| ■ ■ ■ ■ ■ ■ ■ ■ ■ Find an equation of the ellipse. Then find its foci. vertex ͑9, Ϫ1͒ vertex ͑5, 2͒ 1 1 0 1 x x 2 vertices ͑0, Ϯ1͒ foci ͑Ϯ6, 0͒, vertices ͑Ϯ4, 0͒ 45. Hyperbola, y foci ͑0, Ϯ3͒, foci ͑1, 3͒ and ͑7, 3͒, 46. Hyperbola, foci ͑2, Ϫ2͒ and ͑2, 8͒, 47. Hyperbola, 18. y 43. Hyperbola, 44. Hyperbola, 17. vertices ͑Ϯ3, 0͒, vertices ͑2, 3͒ and ͑6, 3͒ vertices ͑2, 0͒ and ͑2, 6͒ asymptotes y ෇ Ϯ2x foci ͑2, 2͒ and ͑6, 2͒, asymptotes y ෇ x Ϫ 2 and y ෇ 6 Ϫ x 48. Hyperbola, ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 49. The point in a lunar orbit nearest the surface of the moon is 19–24 Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph. 19. |||| x2 y2 Ϫ ෇1 144 25 21. y 2 Ϫ x 2 ෇ 4 20. y2 x2 Ϫ ෇1 16 36 22. 9x 2 Ϫ 4y 2 ෇ 36 called perilune and the point farthest from the surface is called apolune. The Apollo 11 spacecraft was placed in an elliptical lunar orbit with perilune altitude 110 km and apolune altitude 314 km (above the moon). Find an equation of this ellipse if the radius of the moon is 1728 km and the center of the moon is at one focus. 5E-11(pp 726-735) 1/18/06 9:35 AM Page 727 SECTION 11.5 CONIC SECTIONS 50. A cross-section of a parabolic reflector is shown in the figure. The bulb is located at the focus and the opening at the focus is 10 cm. (a) Find an equation of the parabola. (b) Find the diameter of the opening CD , 11 cm from the vertex. Խ Խ 727 56. (a) Show that the equation of the tangent line to the parabola y 2 ෇ 4px at the point ͑x 0 , y 0͒ can be written as y 0 y ෇ 2p͑x ϩ x 0͒ (b) What is the x-intercept of this tangent line? Use this fact to draw the tangent line. 57. Use Simpson’s Rule with n ෇ 10 to estimate the length of the C A ellipse x 2 ϩ 4y 2 ෇ 4. 58. The planet Pluto travels in an elliptical orbit around the Sun 5 cm 11 cm F 5 cm V ❙❙❙❙ (at one focus). The length of the major axis is 1.18 ϫ 10 10 km and the length of the minor axis is 1.14 ϫ 10 10 km. Use Simpson’s Rule with n ෇ 10 to estimate the distance traveled by the planet during one complete orbit around the Sun. 59. Let P͑x 1, y1͒ be a point on the ellipse x 2͞a 2 ϩ y 2͞b 2 ෇ 1 with B D 51. In the LORAN (LOng RAnge Navigation) radio navigation system, two radio stations located at A and B transmit simultaneous signals to a ship or an aircraft located at P. The onboard computer converts the time difference in receiving these signals into a distance difference PA Ϫ PB , and this, according to the definition of a hyperbola, locates the ship or aircraft on one branch of a hyperbola (see the figure). Suppose that station B is located 400 mi due east of station A on a coastline. A ship received the signal from B 1200 microseconds (␮s) before it received the signal from A. (a) Assuming that radio signals travel at a speed of 980 ft͞␮s, find an equation of the hyperbola on which the ship lies. (b) If the ship is due north of B, how far off the coastline is the ship? Խ Խ Խ Խ foci F1 and F2 and let ␣ and ␤ be the angles between the lines PF1, PF2 and the ellipse as in the figure. Prove that ␣ ෇ ␤. This explains how whispering galleries and lithotripsy work. Sound coming from one focus is reflected and passes through the other focus. [Hint: Use the formula in Problem 13 on page 220 to show that tan ␣ ෇ tan ␤.] y å P(⁄, ›) ∫ F¡ 0 x F™ ¥ ≈ + b@ =1 a@ 60. Let P͑x 1, y1͒ be a point on the hyperbola x 2͞a 2 Ϫ y 2͞b 2 ෇ 1 P A coastline B with foci F1 and F2 and let ␣ and ␤ be the angles between the lines PF1 , PF2 and the hyperbola as shown in the figure. Prove that ␣ ෇ ␤. (This is the reflection property of the hyperbola. It shows that light aimed at a focus F2 of a hyperbolic mirror is reflected toward the other focus F1 .) y 400 mi sending stations å 52. Use the definition of a hyperbola to derive Equation 6 for a P ∫ hyperbola with foci ͑Ϯc, 0͒ and vertices ͑Ϯa, 0͒. 53. Show that the function defined by the upper branch of the F¡ 0 F™ hyperbola y 2͞a 2 Ϫ x 2͞b 2 ෇ 1 is concave upward. 54. Find an equation for the ellipse with foci ͑1, 1͒ and ͑Ϫ1, Ϫ1͒ and major axis of length 4. 55. Determine the type of curve represented by the equation x2 y2 ϩ ෇1 k k Ϫ 16 in each of the following cases: (a) k Ͼ 16, (b) 0 Ͻ k Ͻ 16, and (c) k Ͻ 0. (d) Show that all the curves in parts (a) and (b) have the same foci, no matter what the value of k is. P F¡ F™ x 5E-11(pp 726-735) 728 ❙❙❙❙ 1/18/06 9:35 AM Page 728 CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES |||| 11.6 Conic Sections in Polar Coordinates In the preceding section we defined the parabola in terms of a focus and directrix, but we defined the ellipse and hyperbola in terms of two foci. In this section we give a more unified treatment of all three types of conic sections in terms of a focus and directrix. Furthermore, if we place the focus at the origin, then a conic section has a simple polar equation. In Chapter 14 we will use the polar equation of an ellipse to derive Kepler’s laws of planetary motion. 1 Theorem Let F be a fixed point (called the focus) and l be a fixed line (called the directrix) in a plane. Let e be a fixed positive number (called the eccentricity). The set of all points P in the plane such that Խ PF Խ ෇ e Խ Pl Խ (The ratio of the distance from F to the distance from l is the constant e.) is a conic section. The conic is (a) an ellipse if e Ͻ 1 (b) a parabola if e ෇ 1 (c) a hyperbola if e Ͼ 1 Խ Խ Խ Խ Proof Notice that if the eccentricity is e ෇ 1, then PF ෇ Pl and so the given condiy l (directrix) P r x=d ¨ F x r cos ¨ tion simply becomes the definition of a parabola as given in Section 11.5. Let us place the focus F at the origin and the directrix parallel to the y-axis and d units to the right. Thus, the directrix has equation x ෇ d and is perpendicular to the polar axis. If the point P has polar coordinates ͑r, ␪ ͒, we see from Figure 1 that Խ PF Խ ෇ r Խ Pl Խ ෇ d Ϫ r cos ␪ Thus, the condition Խ PF Խ ͞ Խ Pl Խ ෇ e, or Խ PF Խ ෇ e Խ Pl Խ, becomes r ෇ e͑d Ϫ r cos ␪ ͒ 2 d C FIGURE 1 If we square both sides of this polar equation and convert to rectangular coordinates, we get x 2 ϩ y 2 ෇ e 2͑d Ϫ x͒2 ෇ e 2͑d 2 Ϫ 2dx ϩ x 2 ͒ ͑1 Ϫ e 2 ͒x 2 ϩ 2de 2x ϩ y 2 ෇ e 2d 2 or After completing the square, we have 3 ͩ xϩ e 2d 1 Ϫ e2 ͪ 2 ϩ y2 e 2d 2 ෇ 1 Ϫ e2 ͑1 Ϫ e 2 ͒2 If e Ͻ 1, we recognize Equation 3 as the equation of an ellipse. In fact, it is of the form ͑x Ϫ h͒2 y2 ϩ 2 ෇1 2 a b 5E-11(pp 726-735) 1/18/06 9:35 AM Page 729 SECTION 11.6 CONIC SECTIONS IN POLAR COORDINATES ❙❙❙❙ 729 where h෇Ϫ 4 x=d directrix x e 2d 2 ͑1 Ϫ e 2 ͒2 b2 ෇ e 2d 2 1 Ϫ e2 c෇ This shows that e 2d ෇ Ϫh 1 Ϫ e2 and confirms that the focus as defined in Theorem 1 means the same as the focus defined in Section 11.5. It also follows from Equations 4 and 5 that the eccentricity is given by c e෇ a ed 1+e cos ¨ (a) r= a2 ෇ In Section 11.5 we found that the foci of an ellipse are at a distance c from the center, where e 4d 2 c2 ෇ a2 Ϫ b2 ෇ 5 ͑1 Ϫ e 2 ͒2 y F e 2d 1 Ϫ e2 y If e Ͼ 1, then 1 Ϫ e 2 Ͻ 0 and we see that Equation 3 represents a hyperbola. Just as we did before, we could rewrite Equation 3 in the form x=_d directrix ͑x Ϫ h͒2 y2 Ϫ 2 ෇1 2 a b F x and see that e෇ (b) r= ed 1-e cos ¨ directrix F x If the directrix is chosen to be to the left of the focus as x ෇ Ϫd, or if the directrix is chosen to be parallel to the polar axis as y ෇ Ϯd, then the polar equation of the conic is given by the following theorem, which is illustrated by Figure 2. (See Exercises 21–23.) 6 (c) r= where c 2 ෇ a 2 ϩ b 2 By solving Equation 2 for r, we see that the polar equation of the conic shown in Figure 1 can be written as ed r෇ 1 ϩ e cos ␪ y y=d c a ed 1+e sin ¨ Theorem A polar equation of the form r෇ y ed 1 Ϯ e cos ␪ or r෇ ed 1 Ϯ e sin ␪ represents a conic section with eccentricity e. The conic is an ellipse if e Ͻ 1, a parabola if e ෇ 1, or a hyperbola if e Ͼ 1. F y=_d (d) r= x directrix ed 1-e sin ¨ FIGURE 2 Polar equations of conics EXAMPLE 1 Find a polar equation for a parabola that has its focus at the origin and whose directrix is the line y ෇ Ϫ6. SOLUTION Using Theorem 6 with e ෇ 1 and d ෇ 6, and using part (d) of Figure 2, we see that the equation of the parabola is r෇ 6 1 Ϫ sin ␪ 5E-11(pp 726-735) 730 ❙❙❙❙ 1/18/06 9:36 AM Page 730 CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES EXAMPLE 2 A conic is given by the polar equation r෇ 10 3 Ϫ 2 cos ␪ Find the eccentricity, identify the conic, locate the directrix, and sketch the conic. SOLUTION Dividing numerator and denominator by 3, we write the equation as r෇ 10 r= 3-2 cos ¨ d෇ focus 0 x (10, 0) FIGURE 3 EXAMPLE 3 Sketch the conic r ෇ 10 3 2 3 ෇5 12 . 2 ϩ 4 sin ␪ π ”6, ’ 2 r෇ π y=3 (directrix) x (6, 0) focus FIGURE 4 r= e ෇ SOLUTION Writing the equation in the form y (6, π) 0 10 3 so the directrix has Cartesian equation x ෇ Ϫ5. When ␪ ෇ 0, r ෇ 10; when ␪ ෇ ␲, r ෇ 2. So the vertices have polar coordinates ͑10, 0͒ and ͑2, ␲͒. The ellipse is sketched in Figure 3. (2, π) ”2, ’ 2 1 Ϫ cos ␪ From Theorem 6 we see that this represents an ellipse with e ෇ 2 . Since ed ෇ 10 , 3 3 we have y x=_5 (directrix) 10 3 2 3 12 2+4 sin ¨ 6 1 ϩ 2 sin ␪ we see that the eccentricity is e ෇ 2 and the equation therefore represents a hyperbola. Since ed ෇ 6, d ෇ 3 and the directrix has equation y ෇ 3. The vertices occur when ␪ ෇ ␲͞2 and 3␲͞2, so they are ͑2, ␲͞2͒ and ͑Ϫ6, 3␲͞2͒ ෇ ͑6, ␲͞2͒. It is also useful to plot the x-intercepts. These occur when ␪ ෇ 0, ␲ ; in both cases r ෇ 6. For additional accuracy we could draw the asymptotes. Note that r l Ϯϱ when 2 ϩ 4 sin ␪ l 0 ϩ or 0 Ϫ and 2 ϩ 4 sin ␪ ෇ 0 when sin ␪ ෇ Ϫ 1 . Thus, the asymptotes are parallel to the rays 2 ␪ ෇ 7␲͞6 and ␪ ෇ 11␲͞6. The hyperbola is sketched in Figure 4. When rotating conic sections, we find it much more convenient to use polar equations than Cartesian equations. We just use the fact (see Exercise 75 in Section 11.3) that the graph of r ෇ f ͑␪ Ϫ ␣͒ is the graph of r ෇ f ͑␪ ͒ rotated counterclockwise about the origin through an angle ␣. EXAMPLE 4 If the ellipse of Example 2 is rotated through an angle ␲͞4 about the origin, find a polar equation and graph the resulting ellipse. 11 10 r=3-2 cos(¨-π/4) _5 15 10 r= 3-2 cos ¨ _6 FIGURE 5 SOLUTION We get the equation of the rotated ellipse by replacing ␪ with ␪ Ϫ ␲͞4 in the equation given in Example 2. So the new equation is r෇ 10 3 Ϫ 2 cos͑␪ Ϫ ␲͞4͒ We use this equation to graph the rotated ellipse in Figure 5. Notice that the ellipse has been rotated about its left focus. 5E-11(pp 726-735) 1/18/06 9:36 AM Page 731 SECTION 11.6 CONIC SECTIONS IN POLAR COORDINATES ❙❙❙❙ 731 In Figure 6 we use a computer to sketch a number of conics to demonstrate the effect of varying the eccentricity e. Notice that when e is close to 0 the ellipse is nearly circular, whereas it becomes more elongated as e l 1Ϫ. When e ෇ 1, of course, the conic is a parabola. e=0.1 e=0.5 e=0.68 e=1 e=1.1 e=0.86 e=0.96 e=1.4 e=4 FIGURE 6 |||| 11.6 Exercises 1–8 |||| Write a polar equation of a conic with the focus at the origin and the given data. 1. Hyperbola, eccentricity 7, 4 eccentricity 3, 4 4. Hyperbola, directrix x ෇ Ϫ5 eccentricity 2, 6. Ellipse, 7. Ellipse, eccentricity 1, 2 8. Hyperbola, 9–16 graph the curve obtained by rotating this parabola about its focus through an angle ␲͞6. ■ ■ and 1.0 on a common screen. How does the value of e affect the shape of the curve? directrix r ෇ 4 sec ␪ eccentricity 3, ■ ; 19. Graph the conics r ෇ e͑͞1 Ϫ e cos ␪ ͒ with e ෇ 0.4, 0.6, 0.8, vertex ͑1, ␲͞2͒ eccentricity 0.8, ■ ; 18. Graph the parabola r ෇ 5͑͞2 ϩ 2 sin ␪ ͒ and its directrix. Also directrix y ෇ Ϫ2 vertex ͑4, 3␲͞2͒ 5. Parabola, ■ r ෇ 1͑͞4 Ϫ 3 cos ␪ ͒ and graph the conic and its directrix. (b) If this conic is rotated counterclockwise about the origin through an angle ␲͞3, write the resulting equation and graph its curve. directrix y ෇ 6 directrix x ෇ 4 2. Parabola, 3. Ellipse, ; 17. (a) Find the eccentricity and directrix of the conic ■ directrix r ෇ Ϫ6 csc ␪ ■ ■ ■ ■ ; 20. (a) Graph the conics r ෇ ed͑͞1 ϩ e sin ␪ ͒ for e ෇ 1 and vari■ ■ (a) Find the eccentricity, (b) identify the conic, (c) give an equation of the directrix, and (d) sketch the conic. |||| 9. r ෇ 1 1 ϩ sin ␪ 10. r ෇ 6 3 ϩ 2 sin ␪ 11. r ෇ 12 4 Ϫ sin ␪ 12. r ෇ 4 2 Ϫ 3 cos ␪ 13. r ෇ 9 6 ϩ 2 cos ␪ 14. r ෇ 5 2 Ϫ 2 sin ␪ 15. r ෇ 3 4 Ϫ 8 cos ␪ 16. r ෇ 4 2 ϩ cos ␪ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ous values of d. How does the value of d affect the shape of the conic? (b) Graph these conics for d ෇ 1 and various values of e. How does the value of e affect the shape of the conic? 21. Show that a conic with focus at the origin, eccentricity e, and directrix x ෇ Ϫd has polar equation r෇ ed 1 Ϫ e cos ␪ 22. Show that a conic with focus at the origin, eccentricity e, and directrix y ෇ d has polar equation r෇ ■ ■ ed 1 ϩ e sin ␪ 5E-11(pp 726-735) ❙❙❙❙ 732 1/18/06 9:36 AM Page 732 CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES 23. Show that a conic with focus at the origin, eccentricity e, and aphelion, respectively. Use Exercise 25(a) to show that the perihelion distance from a planet to the Sun is a͑1 Ϫ e͒ and the aphelion distance is a͑1 ϩ e͒. (b) Use the data of Exercise 25(b) to find the distances from Earth to the Sun at perihelion and at aphelion. directrix y ෇ Ϫd has polar equation r෇ ed 1 Ϫ e sin ␪ 24. Show that the parabolas r ෇ c͑͞1 ϩ cos ␪ ͒ and 27. The orbit of Halley’s comet, last seen in 1986 and due to r ෇ d͑͞1 Ϫ cos ␪ ͒ intersect at right angles. return in 2062, is an ellipse with eccentricity 0.97 and one focus at the Sun. The length of its major axis is 36.18 AU. [An astronomical unit (AU) is the mean distance between Earth and the Sun, about 93 million miles.] Find a polar equation for the orbit of Halley’s comet. What is the maximum distance from the comet to the Sun? 25. (a) Show that the polar equation of an ellipse with directrix x ෇ Ϫd can be written in the form r෇ a͑1 Ϫ e 2 ͒ 1 Ϫ e cos ␪ (b) Find an approximate polar equation for the elliptical orbit of Earth around the Sun (at one focus) given that the eccentricity is about 0.017 and the length of the major axis is about 2.99 ϫ 10 8 km. 26. (a) The planets move around the Sun in elliptical orbits with 28. The Hale-Bopp comet, discovered in 1995, has an elliptical orbit with eccentricity 0.9951 and the length of the major axis is 356.5 AU. Find a polar equation for the orbit of this comet. How close to the Sun does it come? 29. The planet Mercury travels in an elliptical orbit with eccentric- ity 0.206. Its minimum distance from the Sun is 4.6 ϫ 10 7 km. Use the results of Exercise 26(a) to find its maximum distance from the Sun. the Sun at one focus. The positions of a planet that are closest to and farthest from the Sun are called its perihelion and planet 30. The distance from the planet Pluto to the Sun is 4.43 ϫ 10 9 km at perihelion and 7.37 ϫ 10 9 km at aphelion. Use Exercise 26 to find the eccentricity of Pluto’s orbit. r aphelion |||| ¨ Sun 31. Using the data from Exercise 29, find the distance traveled by perihelion 11 Review the planet Mercury during one complete orbit around the Sun. (If your calculator or computer algebra system evaluates definite integrals, use it. Otherwise, use Simpson’s Rule.) ■ CONCEPT CHECK 1. (a) What is a parametric curve? (b) How do you sketch a parametric curve? 2. (a) How do you find the slope of a tangent to a parametric curve? (b) How do you find the area under a parametric curve? 3. Write an expression for each of the following: (a) The length of a parametric curve (b) The area of the surface obtained by rotating a parametric curve about the x-axis 4. (a) Use a diagram to explain the meaning of the polar coordi- nates ͑r, ␪ ͒ of a point. (b) Write equations that express the Cartesian coordinates ͑x, y͒ of a point in terms of the polar coordinates. (c) What equations would you use to find the polar coordinates of a point if you knew the Cartesian coordinates? 5. (a) How do you find the slope of a tangent line to a polar curve? (b) How do you find the area of a region bounded by a polar curve? ■ (c) How do you find the length of a polar curve? 6. (a) Give a geometric definition of a parabola. (b) Write an equation of a parabola with focus ͑0, p͒ and directrix y ෇ Ϫp. What if the focus is ͑ p, 0͒ and the directrix is x ෇ Ϫp? 7. (a) Give a definition of an ellipse in terms of foci. (b) Write an equation for the ellipse with foci ͑Ϯc, 0͒ and vertices ͑Ϯa, 0͒. 8. (a) Give a definition of a hyperbola in terms of foci. (b) Write an equation for the hyperbola with foci ͑Ϯc, 0͒ and vertices ͑Ϯa, 0͒. (c) Write equations for the asymptotes of the hyperbola in part (b). 9. (a) What is the eccentricity of a conic section? (b) What can you say about the eccentricity if the conic section is an ellipse? A hyperbola? A parabola? (c) Write a polar equation for a conic section with eccentricity e and directrix x ෇ d. What if the directrix is x ෇ Ϫd ? y ෇ d ? y ෇ Ϫd ? 5E-11(pp 726-735) 1/18/06 9:37 AM Page 733 CHAPTER 11 REVIEW ■ TRUE-FALSE QUIZ Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 6. The equations r ෇ 2, x 2 ϩ y 2 ෇ 4, and x ෇ 2 sin 3t, y ෇ 2 cos 3t ͑0 ഛ t ഛ 2␲ ͒ all have the same graph. 2. If x ෇ f ͑t͒ and y ෇ t͑t͒ are twice differentiable, then 7. The parametric equations x ෇ t 2, y ෇ t 4 have the same graph d 2 y͞dx 2 ෇ ͑d 2 y͞dt 2 ͒͑͞d 2x͞dt 2 ͒. as x ෇ t 3, y ෇ t 6. 3. The length of the curve x ෇ f ͑t͒, y ෇ t͑t͒, a ഛ t ഛ b, is xabs͓ f Ј͑t͔͒ 2 ϩ ͓ tЈ͑t͔͒ 2 dt . 8. The graph of y 2 ෇ 2y ϩ 3x is a parabola. 4. If a point is represented by ͑x, y͒ in Cartesian coordinates 9. A tangent line to a parabola intersects the parabola only once. (where x 0) and ͑r, ␪ ͒ in polar coordinates, then ␪ ෇ tan Ϫ1͑ y͞x͒. 10. A hyperbola never intersects its directrix. ■ EXERCISES |||| Find a polar equation for the curve represented by the given Cartesian equation. |||| Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. 1. x ෇ t 2 ϩ 4t, y ෇ 2 Ϫ t, Ϫ4 ഛ t ഛ 1 2. x ෇ 1 ϩ e 2t, y ෇ et ■ 15. x ϩ y ෇ 2 ■ y ෇ cot ␪ ■ ■ ■ 16. x 2 ϩ y 2 ෇ 2 ■ ■ ■ ■ ■ ■ ■ ■ ■ metric curve x ෇ f ͑t͒, y ෇ t͑t͒. Indicate with arrows the direction in which the curve is traced as t increases. x y 19. x ෇ ln t, y ෇ 1 ϩ t 2; 20. x ෇ t ϩ 6t ϩ 1, 3 t Ϫ␪ 21. r ෇ e ; t 1 Sketch the polar curve. 8. r ෇ sin 4␪ 9. r ෇ 1 ϩ cos 2␪ 11. r 2 ෇ sec 2␪ 1 13. r ෇ 1 ϩ cos ␪ 8 14. r ෇ 4 ϩ 3 sin ␪ ■ ■ ■ y ෇ 2t Ϫ t 2; ■ t ෇ Ϫ1 ■ ␪ ෇ ␲ ͞2 ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ Find dy͞dx and d 2 y͞dx 2. y ෇ t sin t 24. x ෇ 1 ϩ t 2, 12. r ෇ 2 cos͑␪ ͞ 2͒ ■ ■ |||| t෇1 23. x ෇ t cos t, 10. r ෇ 3 ϩ cos 3␪ ■ ■ 23–24 7. r ෇ 1 Ϫ cos ␪ ■ ■ ␪෇␲ 22. r ෇ 3 ϩ cos 3 ␪ ; ■ ■ ■ |||| Find the slope of the tangent line to the given curve at the point corresponding to the specified value of the parameter. _1 ■ ■ 19–22 1 ■ ■ graph the ellipse obtained by rotation about the origin through an angle 2␲͞3. 6. Use the graphs of x ෇ f ͑t͒ and y ෇ t͑t͒ to sketch the para- |||| ■ ; 18. Graph the ellipse r ෇ 2͑͞4 Ϫ 3 cos ␪ ͒ and its directrix. Also curve y ෇ sx. 7–14 ■ cochleoid. Use a graph of r as a function of ␪ in Cartesian coordinates to sketch the cochleoid by hand. Then graph it with a machine to check your sketch. ■ 5. Write three different sets of parametric equations for the 1 ■ ; 17. The curve with polar equation r ෇ ͑sin ␪ ͒͞␪ is called a y ෇ 1 ϩ sin ␪ ■ ■ 15–16 1–4 ■ ■ same graph. then it has a horizontal tangent when t ෇ 1. 4. x ෇ 2 cos ␪, 733 5. The polar curves r ෇ 1 Ϫ sin 2␪ and r ෇ sin 2␪ Ϫ 1 have the 1. If the parametric curve x ෇ f ͑t͒, y ෇ t͑t͒ satisfies tЈ͑1͒ ෇ 0, 3. x ෇ tan ␪, ❙❙❙❙ y ෇ t Ϫ t3 ■ ■ ■ ■ ■ ■ ; 25. Use a graph to estimate the coordinates of the lowest point on ■ ■ the curve x ෇ t 3 Ϫ 3t, y ෇ t 2 ϩ t ϩ 1. Then use calculus to find the exact coordinates. 5E-11(pp 726-735) 734 ❙❙❙❙ 1/18/06 9:37 AM Page 734 CHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES 26. Find the area enclosed by the loop of the curve in Exercise 25. 43. x ෇ 2a cos t Ϫ a cos 2t y ෇ 2a sin t Ϫ a sin 2t 46. 25x 2 ϩ 4y 2 ϩ 50x Ϫ 16y ෇ 59 29. Find the area enclosed by the curve r 2 ෇ 9 cos 5␪. ■ 30. Find the area enclosed by the inner loop of the curve ■ ■ ■ ■ ■ ■ asymptotes 2y ෇ Ϯx. 33. Find the area of the region that lies inside both of the circles 50. Find an equation of the ellipse with foci ͑3, Ϯ2͒ and major axis r ෇ 2 sin ␪ and r ෇ sin ␪ ϩ cos ␪. with length 8. 34. Find the area of the region that lies inside the curve 51. Find an equation for the ellipse that shares a vertex and a focus r ෇ 2 ϩ cos 2␪ but outside the curve r ෇ 2 ϩ sin ␪. with the parabola x 2 ϩ y ෇ 100 and that has its other focus at the origin. Find the length of the curve. y ෇ 2t 3, 52. Show that if m is any real number, then there are exactly 0ഛtഛ2 y ෇ cosh 3t, two lines of slope m that are tangent to the ellipse x 2͞a 2 ϩ y 2͞b 2 ෇ 1 and their equations are y ෇ mx Ϯ sa 2m 2 ϩ b 2. 0ഛtഛ1 ␲ ഛ ␪ ഛ 2␲ 38. r ෇ sin ͑␪͞3͒, 53. Find a polar equation for the ellipse with focus at the origin, 0ഛ␪ഛ␲ 3 ■ ■ ■ eccentricity 1 , and directrix with equation r ෇ 4 sec ␪. 3 ■ ■ ■ ■ ■ ■ 54. Show that the angles between the polar axis and the 39–40 Find the area of the surface obtained by rotating the given curve about the x-axis. |||| 39. x ෇ 4 st, ■ ■ ■ t3 1 ϩ 2, 3 2t y ෇ cosh 3t, 0ഛtഛ1 ■ ■ ■ ■ ■ asymptotes of the hyperbola r ෇ ed͑͞1 Ϫ e cos ␪ ͒, e Ͼ 1, are given by cosϪ1͑Ϯ1͞e͒. 55. In the figure the circle of radius a is stationary, and for every 1ഛtഛ4 y෇ 40. x ෇ 2 ϩ 3t, ■ 49. Find an equation of the hyperbola with foci ͑Ϯ3, 0͒ and r ෇ 2 cos ␪. ■ ■ vertices ͑0, Ϯ2͒. 32. Find the points of intersection of the curves r ෇ cot ␪ and ■ ■ 48. Find an equation of the hyperbola with foci ͑0, Ϯ5͒ and r ෇ 4 cos ␪. ■ ■ y ෇ 2. 31. Find the points of intersection of the curves r ෇ 2 and 37. r ෇ 1͞␪, ■ 47. Find an equation of the parabola with focus ͑0, 6͒ and directrix r ෇ 1 Ϫ 3 sin ␪. 36. x ෇ 2 ϩ 3t, x y2 ϩ ෇1 9 8 45. 6y 2 ϩ x Ϫ 36y ϩ 55 ෇ 0 28. Find the area enclosed by the curve in Exercise 27. 35. x ෇ 3t 2, Find the foci and vertices and sketch the graph. 44. 4x 2 Ϫ y 2 ෇ 16 have vertical or horizontal tangents? Use this information to help sketch the curve. |||| |||| 2 27. At what points does the curve 35–38 43–46 ␪, the point P is the midpoint of the segment QR. The curve traced out by P for 0 Ͻ ␪ Ͻ ␲ is called the longbow curve. Find parametric equations for this curve. ■ ■ ■ ■ y ; 41. The curves defined by the parametric equations x෇ t2 Ϫ c t2 ϩ 1 y෇ R 2a t͑t 2 Ϫ c͒ t2 ϩ 1 are called strophoids (from a Greek word meaning “to turn or twist”). Investigate how these curves vary as c varies. P a a ; 42. A family of curves has polar equations r ෇ Խ sin 2␪ Խ where a is a positive number. Investigate how the curves change as a changes. y=2 a Q ¨ 0 x 5E-11(pp 726-735) 1/18/06 9:37 AM Page 735 PROBLEMS PLUS 1. A curve is defined by the parametric equations x෇y t 1 cos u du u y෇y t 1 sin u du u Find the length of the arc of the curve from the origin to the nearest point where there is a vertical tangent line. 2. (a) Find the highest and lowest points on the curve x 4 ϩ y 4 ෇ x 2 ϩ y 2. CAS (b) Sketch the curve. (Notice that it is symmetric with respect to both axes and both of the lines y ෇ Ϯx, so it suffices to consider y ജ x ജ 0 initially.) (c) Use polar coordinates and a computer algebra system to find the area enclosed by the curve. ; 3. What is the smallest viewing rectangle that contains every member of the family of polar curves r ෇ 1 ϩ c sin ␪, where 0 ഛ c ഛ 1? Illustrate your answer by graphing several members of the family in this viewing rectangle. 4. Four bugs are placed at the four corners of a square with side length a. The bugs crawl counterclockwise at the same speed and each bug crawls directly toward the next bug at all times. They approach the center of the square along spiral paths. (a) Find the polar equation of a bug’s path assuming the pole is at the center of the square. (Use the fact that the line joining one bug to the next is tangent to the bug’s path.) (b) Find the distance traveled by a bug by the time it meets the other bugs at the center. a a a a 5. A curve called the folium of Descartes is defined by the parametric equations x෇ 3t 1 ϩ t3 y෇ 3t 2 1 ϩ t3 (a) Show that if ͑a, b͒ lies on the curve, then so does ͑b, a͒; that is, the curve is symmetric with respect to the line y ෇ x. Where does the curve intersect this line? (b) Find the points on the curve where the tangent lines are horizontal or vertical. (c) Show that the line y ෇ Ϫx Ϫ 1 is a slant asymptote. (d) Sketch the curve. (e) Show that a Cartesian equation of this curve is x 3 ϩ y 3 ෇ 3xy. (f) Show that the polar equation can be written in the form r෇ CAS 3 sec ␪ tan ␪ 1 ϩ tan 3␪ (g) Find the area enclosed by the loop of this curve. (h) Show that the area of the loop is the same as the area that lies between the asymptote and the infinite branches of the curve. (Use a computer algebra system to evaluate the integral.) 735 ...
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This note was uploaded on 02/08/2010 for the course M 340L taught by Professor Lay during the Spring '10 term at École Normale Supérieure.

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