Solutions for Conic Sections in Polar Coordinates

Solutions to Try Its

1. ellipse; [latex]e=\frac{1}{3};x=-2[/latex]

2.

3. [latex]r=\frac{1}{1-\cos \theta }[/latex]

4. [latex]4 - 8x+3{x}^{2}-{y}^{2}=0[/latex]

Solutions to Odd-Numbered Exercises

1. If eccentricity is less than 1, it is an ellipse. If eccentricity is equal to 1, it is a parabola. If eccentricity is greater than 1, it is a hyperbola.

3. The directrix will be parallel to the polar axis.

5. One of the foci will be located at the origin.

7. Parabola with [latex]e=1[/latex] and directrix [latex]\frac{3}{4}[/latex] units below the pole.

9. Hyperbola with [latex]e=2[/latex] and directrix [latex]\frac{5}{2}[/latex] units above the pole.

11. Parabola with [latex]e=1[/latex] and directrix [latex]\frac{3}{10}[/latex] units to the right of the pole.

13. Ellipse with [latex]e=\frac{2}{7}[/latex] and directrix [latex]2[/latex] units to the right of the pole.

15. Hyperbola with [latex]e=\frac{5}{3}[/latex] and directrix [latex]\frac{11}{5}[/latex] units above the pole.

17. Hyperbola with [latex]e=\frac{8}{7}[/latex] and directrix [latex]\frac{7}{8}[/latex] units to the right of the pole.

19. [latex]25{x}^{2}+16{y}^{2}-12y - 4=0[/latex]

21. [latex]21{x}^{2}-4{y}^{2}-30x+9=0[/latex]

23. [latex]64{y}^{2}=48x+9[/latex]

25. [latex]96{y}^{2}-25{x}^{2}+110y+25=0[/latex]

27. [latex]3{x}^{2}+4{y}^{2}-2x - 1=0[/latex]

29. [latex]5{x}^{2}+9{y}^{2}-24x - 36=0[/latex]

31.

33.

35.

37.

39.

41.

43. [latex]r=\frac{4}{5+\cos \theta }[/latex]

45. [latex]r=\frac{4}{1+2\sin \theta }[/latex]

47. [latex]r=\frac{1}{1+\cos \theta }[/latex]

49. [latex]r=\frac{7}{8 - 28\cos \theta }[/latex]

51. [latex]r=\frac{12}{2+3\sin \theta }[/latex]

53. [latex]r=\frac{15}{4 - 3\cos \theta }[/latex]

55. [latex]r=\frac{3}{3 - 3\cos \theta }[/latex]

57. [latex]r=\pm \frac{2}{\sqrt{1+\sin \theta \cos \theta }}[/latex]

59. [latex]r=\pm \frac{2}{4\cos \theta +3\sin \theta }[/latex]