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 $\sin \text{ }\theta$, 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. $r=\frac{6}{1 - 2\text{ }\cos \text{ }\theta }$

7. $r=\frac{3}{4 - 4\text{ }\sin \text{ }\theta }$

8. $r=\frac{8}{4 - 3\text{ }\cos \text{ }\theta }$

9. $r=\frac{5}{1+2\text{ }\sin \text{ }\theta }$

10. $r=\frac{16}{4+3\text{ }\cos \text{ }\theta }$

11. $r=\frac{3}{10+10\text{ }\cos \text{ }\theta }$

12. $r=\frac{2}{1-\cos \text{ }\theta }$

13. $r=\frac{4}{7+2\text{ }\cos \text{ }\theta }$

14. $r\left(1-\cos \text{ }\theta \right)=3$

15. $r\left(3+5\sin \text{ }\theta \right)=11$

16. $r\left(4 - 5\sin \text{ }\theta \right)=1$

17. $r\left(7+8\cos \text{ }\theta \right)=7$

For the following exercises, convert the polar equation of a conic section to a rectangular equation.

18. $r=\frac{4}{1+3\text{ }\sin \text{ }\theta }$

19. $r=\frac{2}{5 - 3\text{ }\sin \text{ }\theta }$

20. $r=\frac{8}{3 - 2\text{ }\cos \text{ }\theta }$

21. $r=\frac{3}{2+5\text{ }\cos \text{ }\theta }$

22. $r=\frac{4}{2+2\text{ }\sin \text{ }\theta }$

23. $r=\frac{3}{8 - 8\text{ }\cos \text{ }\theta }$

24. $r=\frac{2}{6+7\text{ }\cos \text{ }\theta }$

25. $r=\frac{5}{5 - 11\text{ }\sin \text{ }\theta }$

26. $r\left(5+2\text{ }\cos \text{ }\theta \right)=6$

27. $r\left(2-\cos \text{ }\theta \right)=1$

28. $r\left(2.5 - 2.5\text{ }\sin \text{ }\theta \right)=5$

29. $r=\frac{6\sec \text{ }\theta }{-2+3\text{ }\sec \text{ }\theta }$

30. $r=\frac{6\csc \text{ }\theta }{3+2\text{ }\csc \text{ }\theta }$

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. $r=\frac{5}{2+\cos \text{ }\theta }$

32. $r=\frac{2}{3+3\text{ }\sin \text{ }\theta }$

33. $r=\frac{10}{5 - 4\text{ }\sin \text{ }\theta }$

34. $r=\frac{3}{1+2\text{ }\cos \text{ }\theta }$

35. $r=\frac{8}{4 - 5\text{ }\cos \text{ }\theta }$

36. $r=\frac{3}{4 - 4\text{ }\cos \text{ }\theta }$

37. $r=\frac{2}{1-\sin \text{ }\theta }$

38. $r=\frac{6}{3+2\text{ }\sin \text{ }\theta }$

39. $r\left(1+\cos \text{ }\theta \right)=5$

40. $r\left(3 - 4\sin \text{ }\theta \right)=9$

41. $r\left(3 - 2\sin \text{ }\theta \right)=6$

42. $r\left(6 - 4\cos \text{ }\theta \right)=5$

For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.

43. Directrix: $x=4;e=\frac{1}{5}$

44. Directrix: $x=-4;e=5$

45. Directrix: $y=2;e=2$

46. Directrix: $y=-2;e=\frac{1}{2}$

47. Directrix: $x=1;e=1$

48. Directrix: $x=-1;e=1$

49. Directrix: $x=-\frac{1}{4};e=\frac{7}{2}$

50. Directrix: $y=\frac{2}{5};e=\frac{7}{2}$

51. Directrix: $y=4;e=\frac{3}{2}$

52. Directrix: $x=-2;e=\frac{8}{3}$

53. Directrix: $x=-5;e=\frac{3}{4}$

54. Directrix: $y=2;e=2.5$

55. Directrix: $x=-3;e=\frac{1}{3}$

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 $\theta$.

56. $xy=2$

57. ${x}^{2}+xy+{y}^{2}=4$

58. $2{x}^{2}+4xy+2{y}^{2}=9$

59. $16{x}^{2}+24xy+9{y}^{2}=4$

60. $2xy+y=1$