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 [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]

Licenses & Attributions