SalasSV_09_03_ex - 538 CHAPTER 9 THE CONIC SECTIONS; POLAR...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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

This note was uploaded on 10/12/2010 for the course MATH 12345 taught by Professor Smith during the Spring '10 term at University of Houston - Downtown.

Ask a homework question - tutors are online