*This preview shows
page 1. Sign up
to
view the full content.*

**Unformatted text preview: **538 CHAPTER 9 THE CONIC SECTIONS; POLAR COORDINATES; PARAMETRIC EQUATIONS
y [r, π – θ ] y [r, θ ] y [r, θ ] [r, θ ] x [r, –θ ] x x [r, π + θ ] symmetry about the y-axis symmetry about the origin symmetry about the x-axis Figure 9.3.11 Example 5 Test the curve r 2 = cos 2θ for symmetry.
SOLUTION Since cos [2(−θ )] = cos (−2θ ) = cos 2θ , you can see that, if [r , θ ] is on the curve, then so is [r , −θ ]. This says that the curve is symmetric about the x-axis. Since cos [2(π − θ )] = cos (2π − 2θ ) = cos (−2θ ) = cos 2θ , you can see that, if [r , θ ] is on the curve, then so is [r , π − θ ]. The curve is therefore symmetric about the y-axis. Being symmetric about both axes, the curve must also be symmetric about the origin. You can verify this directly by noting that cos [2(π + θ )] = cos (2π + 2θ ) = cos 2θ , so that, if [r , θ ] lies on the curve, then so does [r , π + θ ]. A sketch of the curve, which is called a lemniscate, appears in Figure 9.3.12.
y (–1, 0) [1, π ] (1, 0) [1, 0] x r 2 = cos 2θ lemniscate Figure 9.3.12 EXERCISES 9.3
Plot the point given in polar coordinates. 1. 3. 5. 7. [1, 1 π ]. 3 [−1, 1 π ]. 3 [4, 5 π ]. 4 [− 1 , π ]. 2
1 π ]. 2 13. [−3, − 1 π ]. 3 15. [3, − 1 π ]. 2 14. [2, 0]. 16. [2, 3π ]. 2. 4. [1, 1 π ]. 2 [−1, − 1 π ]. 3 6. [−2, 0]. 8. [ 1 , 2 π ]. 33 10. 12. [4, 1 π ]. 6 [−1, 1 π ]. 4 Points are speciﬁed in rectangular coordinates. Give all possible polar coordinates for each point. 17. (0, 1). 19. (−3, 0). 21. (2, −2). √ 23. (4 3, 4). 18. (1, 0). √ 22. (3, −3 3). √ 24. ( 3, −1). 20. (4, 4). Find the rectangular coordinates of the point. 9. [3, 11. [−1, −π ]. 9.4 GRAPHING IN POLAR COORDINATES 539 25. Find a formula for the distance between [r1 , θ1 ] and [r2 , θ2 ]. 26. Show that for r1 > 0, r2 > 0, |θ1 − θ2 | < π the distance formula you found in Exercise 25 reduces to the law of cosines. Find the point [r , θ ] symmetric to the given point about: (a) the x-axis; (b) the y-axis; (c) the origin. Express your answer with r > 0 and θ ∈ [0, 2π ). 27. [ 1 , 1 π ]. 26 29. [−2, 1 π ]. 3 28. [3, − 5 π ]. 4 30. [−3, − 7 π ]. 4 57. tan θ = 2. 58. r = 2 sin θ . Write the equation in rectangular coordinates and identify the curve. 6 4 . 60. r = . 59. r = 2 − cos θ 1 + 2 sin θ 2 4 . 62. r = . 61. r = 1 − cos θ 3 + 2 sin θ 63. Show that if a and b are not both zero, then the curve r = a sin θ + b cos θ is a circle. Find the center and the radius. 64. Find a polar equation for the set of points P [r , θ ] such that the distance from P to the pole equals the distance from P to the line x = −d . See the ﬁgure.
y P [r, θ ] x = –d Test the curve for symmetry about the coordinate axes and for symmetry about the origin. 31. r = 2 + cos θ . 33. r ( sin θ + cos θ ) = 1. 35. r 2 sin 2θ = 1. x = 2. 2xy = 1. x2 + ( y − 2)2 = 4. y = x. 32. r = cos 2θ . 34. r sin θ = 1. 36. r 2 cos 2θ = 1. y = 3. x2 + y2 = 9. (x − a)2 + y2 = a2 . x2 − y2 = 4. Write the equation in polar coordinates. 37. 39. 41. 43. 38. 40. 42. 44. 45. x2 + y2 + x = x2 + y2 . 46. y = mx. 48. (x2 + y2 )2 = x2 − y2 . 47. (x2 + y2 )2 = 2xy. Polar axis x Identify the curve and write the equation in rectangular coordinates. 49. r sin θ = 4. 51. θ = 1 π . 3 53. r = 2(1 − cos θ )−1 . 55. r = 3 cos θ . 50. r cos θ = 4. 52. θ 2 = 1 π 2 . 9 54. r = 4 sin (θ + π ). 56. θ = − 1 π . 2 65. Find a polar equation for the set of points P [r , θ ] such that the distance from P to the pole is half the distance from P to the line x = −d . 66. Find a polar equation for the set of points P [r , θ ] such that the distance from P to the pole is twice the distance from P to the line x = −d . 9.4 GRAPHING IN POLAR COORDINATES
We begin with the curve r = θ, θ ≥ 0. The graph is a nonending spiral, part of the famous spiral of Archimedes. The curve is shown in detail from θ = 0 to θ = 2π in Figure 9.4.1. At θ = 0, r = 0; at θ = 1 π , r = 1 π ; at θ = 1 π , r = 1 π ; and so on. 4 4 2 2 The next examples involve trigonometric functions.
Example 1
SOLUTION Sketch the curve r = 1 − 2 cos θ . Since the cosine function is periodic with period 2π , the curve r = 1 − 2 cos θ is a closed curve. We will draw it from θ = 0 to θ = 2π . The curve just repeats itself for values of θ outside the interval [0, 2π ]. ...

View
Full
Document