Section Exercises
1. Explain how eccentricity determines which conic section is given.
2. If a conic section is written as a polar equation, what must be true of the denominator?
3. If a conic section is written as a polar equation, and the denominator involves sin θ, what conclusion can be drawn about the directrix?
4. If the directrix of a conic section is perpendicular to the polar axis, what do we know about the equation of the graph?
5. What do we know about the focus/foci of a conic section if it is written as a polar equation?
For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity.
6. r=1−2 cos θ6
7. r=4−4 sin θ3
8. r=4−3 cos θ8
9. r=1+2 sin θ5
10. r=4+3 cos θ16
11. r=10+10 cos θ3
12. r=1−cos θ2
13. r=7+2 cos θ4
14. r(1−cos θ)=3
15. r(3+5sin θ)=11
16. r(4−5sin θ)=1
17. r(7+8cos θ)=7
For the following exercises, convert the polar equation of a conic section to a rectangular equation.
18. r=1+3 sin θ4
19. r=5−3 sin θ2
20. r=3−2 cos θ8
21. r=2+5 cos θ3
22. r=2+2 sin θ4
23. r=8−8 cos θ3
24. r=6+7 cos θ2
25. r=5−11 sin θ5
26. r(5+2 cos θ)=6
27. r(2−cos θ)=1
28. r(2.5−2.5 sin θ)=5
29. r=−2+3 sec θ6sec θ
30. r=3+2 csc θ6csc θ
For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci.
31. r=2+cos θ5
32. r=3+3 sin θ2
33. r=5−4 sin θ10
34. r=1+2 cos θ3
35. r=4−5 cos θ8
36. r=4−4 cos θ3
37. r=1−sin θ2
38. r=3+2 sin θ6
39. r(1+cos θ)=5
40. r(3−4sin θ)=9
41. r(3−2sin θ)=6
42. r(6−4cos θ)=5
For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.
43. Directrix: x=4;e=51
44. Directrix: x=−4;e=5
45. Directrix: y=2;e=2
46. Directrix: y=−2;e=21
47. Directrix: x=1;e=1
48. Directrix: x=−1;e=1
49. Directrix: x=−41;e=27
50. Directrix: y=52;e=27
51. Directrix: y=4;e=23
52. Directrix: x=−2;e=38
53. Directrix: x=−5;e=43
54. Directrix: y=2;e=2.5
55. Directrix: x=−3;e=31
Equations of conics with an xy term have rotated graphs. For the following exercises, express each equation in polar form with r as a function of θ.
56. xy=2
57. x2+xy+y2=4
58. 2x2+4xy+2y2=9
59. 16x2+24xy+9y2=4
60. 2xy+y=1
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