Problem Set 69: Conic Sections in Polar Coordinates

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]