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 [latex]\sin \text{ }\theta [/latex], 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. [latex]r=\frac{6}{1 - 2\text{ }\cos \text{ }\theta }[/latex]
7. [latex]r=\frac{3}{4 - 4\text{ }\sin \text{ }\theta }[/latex]
8. [latex]r=\frac{8}{4 - 3\text{ }\cos \text{ }\theta }[/latex]
9. [latex]r=\frac{5}{1+2\text{ }\sin \text{ }\theta }[/latex]
10. [latex]r=\frac{16}{4+3\text{ }\cos \text{ }\theta }[/latex]
11. [latex]r=\frac{3}{10+10\text{ }\cos \text{ }\theta }[/latex]
12. [latex]r=\frac{2}{1-\cos \text{ }\theta }[/latex]
13. [latex]r=\frac{4}{7+2\text{ }\cos \text{ }\theta }[/latex]
14. [latex]r\left(1-\cos \text{ }\theta \right)=3[/latex]
15. [latex]r\left(3+5\sin \text{ }\theta \right)=11[/latex]
16. [latex]r\left(4 - 5\sin \text{ }\theta \right)=1[/latex]
17. [latex]r\left(7+8\cos \text{ }\theta \right)=7[/latex]
For the following exercises, convert the polar equation of a conic section to a rectangular equation.
18. [latex]r=\frac{4}{1+3\text{ }\sin \text{ }\theta }[/latex]
19. [latex]r=\frac{2}{5 - 3\text{ }\sin \text{ }\theta }[/latex]
20. [latex]r=\frac{8}{3 - 2\text{ }\cos \text{ }\theta }[/latex]
21. [latex]r=\frac{3}{2+5\text{ }\cos \text{ }\theta }[/latex]
22. [latex]r=\frac{4}{2+2\text{ }\sin \text{ }\theta }[/latex]
23. [latex]r=\frac{3}{8 - 8\text{ }\cos \text{ }\theta }[/latex]
24. [latex]r=\frac{2}{6+7\text{ }\cos \text{ }\theta }[/latex]
25. [latex]r=\frac{5}{5 - 11\text{ }\sin \text{ }\theta }[/latex]
26. [latex]r\left(5+2\text{ }\cos \text{ }\theta \right)=6[/latex]
27. [latex]r\left(2-\cos \text{ }\theta \right)=1[/latex]
28. [latex]r\left(2.5 - 2.5\text{ }\sin \text{ }\theta \right)=5[/latex]
29. [latex]r=\frac{6\sec \text{ }\theta }{-2+3\text{ }\sec \text{ }\theta }[/latex]
30. [latex]r=\frac{6\csc \text{ }\theta }{3+2\text{ }\csc \text{ }\theta }[/latex]
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. [latex]r=\frac{5}{2+\cos \text{ }\theta }[/latex]
32. [latex]r=\frac{2}{3+3\text{ }\sin \text{ }\theta }[/latex]
33. [latex]r=\frac{10}{5 - 4\text{ }\sin \text{ }\theta }[/latex]
34. [latex]r=\frac{3}{1+2\text{ }\cos \text{ }\theta }[/latex]
35. [latex]r=\frac{8}{4 - 5\text{ }\cos \text{ }\theta }[/latex]
36. [latex]r=\frac{3}{4 - 4\text{ }\cos \text{ }\theta }[/latex]
37. [latex]r=\frac{2}{1-\sin \text{ }\theta }[/latex]
38. [latex]r=\frac{6}{3+2\text{ }\sin \text{ }\theta }[/latex]
39. [latex]r\left(1+\cos \text{ }\theta \right)=5[/latex]
40. [latex]r\left(3 - 4\sin \text{ }\theta \right)=9[/latex]
41. [latex]r\left(3 - 2\sin \text{ }\theta \right)=6[/latex]
42. [latex]r\left(6 - 4\cos \text{ }\theta \right)=5[/latex]
For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.
43. Directrix: [latex]x=4;e=\frac{1}{5}[/latex]
44. Directrix: [latex]x=-4;e=5[/latex]
45. Directrix: [latex]y=2;e=2[/latex]
46. Directrix: [latex]y=-2;e=\frac{1}{2}[/latex]
47. Directrix: [latex]x=1;e=1[/latex]
48. Directrix: [latex]x=-1;e=1[/latex]
49. Directrix: [latex]x=-\frac{1}{4};e=\frac{7}{2}[/latex]
50. Directrix: [latex]y=\frac{2}{5};e=\frac{7}{2}[/latex]
51. Directrix: [latex]y=4;e=\frac{3}{2}[/latex]
52. Directrix: [latex]x=-2;e=\frac{8}{3}[/latex]
53. Directrix: [latex]x=-5;e=\frac{3}{4}[/latex]
54. Directrix: [latex]y=2;e=2.5[/latex]
55. Directrix: [latex]x=-3;e=\frac{1}{3}[/latex]
Equations of conics with an [latex]xy[/latex] term have rotated graphs. For the following exercises, express each equation in polar form with [latex]r[/latex] as a function of [latex]\theta [/latex].
56. [latex]xy=2[/latex]
57. [latex]{x}^{2}+xy+{y}^{2}=4[/latex]
58. [latex]2{x}^{2}+4xy+2{y}^{2}=9[/latex]
59. [latex]16{x}^{2}+24xy+9{y}^{2}=4[/latex]
60. [latex]2xy+y=1[/latex]