12.5E: Conic Sections in Polar Coordinates (Exercises)
For the following exercises, given the polar equation of the conic with focus at the origin, identify the eccentricity and directrix.
41. \(r=\frac{10}{1-5 \cos \theta}\)
42. \(r=\frac{6}{3+2 \cos \theta}\)
43. \(r=\frac{1}{4+3 \quad \sin \theta}\)
44. \(r=\frac{3}{5-5} \sin \theta\)
For the following exercises, graph the conic given in polar form. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse or a hyperbola, label the vertices and foci.
45. \(r=\frac{3}{1-\sin \theta}\)
46. \(r=\frac{8}{4+3} \sin \theta\)
47. \(r=\frac{10}{4+5 \cos \theta}\)
48. \(r=\frac{9}{3-6 \quad \cos \theta}\)
For the following exercises, given information about the graph of a conic with focus at the origin, find the equation in polar form.
49. Directrix is \(x=3\) and eccentricity \(e=1\)
50. Directrix is \(y=-2\) and eccentricity \(e=4\)