12.5E: Conic Sections in Polar Coordinates (Exercises)

Last updated
Save as PDF

Page ID: 56135

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

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 \(e=4\)