We've updated our
Privacy Policy effective December 15. Please read our updated Privacy Policy and tap

Study Guides > College Algebra

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\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=612 cos θr=\frac{6}{1 - 2\text{ }\cos \text{ }\theta } 7. r=344 sin θr=\frac{3}{4 - 4\text{ }\sin \text{ }\theta } 8. r=843 cos θr=\frac{8}{4 - 3\text{ }\cos \text{ }\theta } 9. r=51+2 sin θr=\frac{5}{1+2\text{ }\sin \text{ }\theta } 10. r=164+3 cos θr=\frac{16}{4+3\text{ }\cos \text{ }\theta } 11. r=310+10 cos θr=\frac{3}{10+10\text{ }\cos \text{ }\theta } 12. r=21cos θr=\frac{2}{1-\cos \text{ }\theta } 13. r=47+2 cos θr=\frac{4}{7+2\text{ }\cos \text{ }\theta } 14. r(1cos θ)=3r\left(1-\cos \text{ }\theta \right)=3 15. r(3+5sin θ)=11r\left(3+5\sin \text{ }\theta \right)=11 16. r(45sin θ)=1r\left(4 - 5\sin \text{ }\theta \right)=1 17. r(7+8cos θ)=7r\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=41+3 sin θr=\frac{4}{1+3\text{ }\sin \text{ }\theta } 19. r=253 sin θr=\frac{2}{5 - 3\text{ }\sin \text{ }\theta } 20. r=832 cos θr=\frac{8}{3 - 2\text{ }\cos \text{ }\theta } 21. r=32+5 cos θr=\frac{3}{2+5\text{ }\cos \text{ }\theta } 22. r=42+2 sin θr=\frac{4}{2+2\text{ }\sin \text{ }\theta } 23. r=388 cos θr=\frac{3}{8 - 8\text{ }\cos \text{ }\theta } 24. r=26+7 cos θr=\frac{2}{6+7\text{ }\cos \text{ }\theta } 25. r=5511 sin θr=\frac{5}{5 - 11\text{ }\sin \text{ }\theta } 26. r(5+2 cos θ)=6r\left(5+2\text{ }\cos \text{ }\theta \right)=6 27. r(2cos θ)=1r\left(2-\cos \text{ }\theta \right)=1 28. r(2.52.5 sin θ)=5r\left(2.5 - 2.5\text{ }\sin \text{ }\theta \right)=5 29. r=6sec θ2+3 sec θr=\frac{6\sec \text{ }\theta }{-2+3\text{ }\sec \text{ }\theta } 30. r=6csc θ3+2 csc θr=\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=52+cos θr=\frac{5}{2+\cos \text{ }\theta } 32. r=23+3 sin θr=\frac{2}{3+3\text{ }\sin \text{ }\theta } 33. r=1054 sin θr=\frac{10}{5 - 4\text{ }\sin \text{ }\theta } 34. r=31+2 cos θr=\frac{3}{1+2\text{ }\cos \text{ }\theta } 35. r=845 cos θr=\frac{8}{4 - 5\text{ }\cos \text{ }\theta } 36. r=344 cos θr=\frac{3}{4 - 4\text{ }\cos \text{ }\theta } 37. r=21sin θr=\frac{2}{1-\sin \text{ }\theta } 38. r=63+2 sin θr=\frac{6}{3+2\text{ }\sin \text{ }\theta } 39. r(1+cos θ)=5r\left(1+\cos \text{ }\theta \right)=5 40. r(34sin θ)=9r\left(3 - 4\sin \text{ }\theta \right)=9 41. r(32sin θ)=6r\left(3 - 2\sin \text{ }\theta \right)=6 42. r(64cos θ)=5r\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=15x=4;e=\frac{1}{5} 44. Directrix: x=4;e=5x=-4;e=5 45. Directrix: y=2;e=2y=2;e=2 46. Directrix: y=2;e=12y=-2;e=\frac{1}{2} 47. Directrix: x=1;e=1x=1;e=1 48. Directrix: x=1;e=1x=-1;e=1 49. Directrix: x=14;e=72x=-\frac{1}{4};e=\frac{7}{2} 50. Directrix: y=25;e=72y=\frac{2}{5};e=\frac{7}{2} 51. Directrix: y=4;e=32y=4;e=\frac{3}{2} 52. Directrix: x=2;e=83x=-2;e=\frac{8}{3} 53. Directrix: x=5;e=34x=-5;e=\frac{3}{4} 54. Directrix: y=2;e=2.5y=2;e=2.5 55. Directrix: x=3;e=13x=-3;e=\frac{1}{3} Equations of conics with an xyxy term have rotated graphs. For the following exercises, express each equation in polar form with rr as a function of θ\theta . 56. xy=2xy=2 57. x2+xy+y2=4{x}^{2}+xy+{y}^{2}=4 58. 2x2+4xy+2y2=92{x}^{2}+4xy+2{y}^{2}=9 59. 16x2+24xy+9y2=416{x}^{2}+24xy+9{y}^{2}=4 60. 2xy+y=12xy+y=1

Licenses & Attributions