Chapter 8

Plot the point with the given polar coordinates. $$ (3, \pi) $$

Determine the eccentricity, identify the conic, and sketch its graph. $$ r=\frac{2}{1-\sin \theta} $$

Fill in the table for the given set of parametric equations. Find the \(x\) - and \(y\) -intercepts. Sketch the curve and indicate its orientation. $$ x=t+2, y=3+\frac{1}{2} t, \quad-\infty

Plot the point with the given polar coordinates. $$ (2,-\pi / 2) $$

Determine the eccentricity, identify the conic, and sketch its graph. $$ r=\frac{2}{2-\cos \theta} $$

Fill in the table for the given set of parametric equations. Find the \(x\) - and \(y\) -intercepts. Sketch the curve and indicate its orientation. $$ x=2 t+1, y=t^{2}+t,-\infty

Plot the point with the given polar coordinates. $$ \left(-\frac{1}{2}, \pi / 2\right) $$

Determine the eccentricity, identify the conic, and sketch its graph. $$ r=\frac{16}{4+\cos \theta} $$

Sketch the curve that has the given set of parametric equations. $$ x=t-1, y=2 t-1,-1 \leq t \leq 5 $$

Plot the point with the given polar coordinates. $$ (-1, \pi / 6) $$

Determine the eccentricity, identify the conic, and sketch its graph. $$ r=\frac{5}{2+2 \sin \theta} $$

Sketch the curve that has the given set of parametric equations. $$ x=t^{2}-1, y=3 t,-2 \leq t \leq 3 $$

Plot the point with the given polar coordinates. $$ (-4,-\pi / 6) $$

Determine the eccentricity, identify the conic, and sketch its graph. $$ r=\frac{4}{1+2 \sin \theta} $$

Sketch the curve that has the given set of parametric equations. $$ x=\sqrt{t}, y=5-t, \quad t \geq 0 $$

Plot the point with the given polar coordinates. $$ \left(\frac{2}{3}, 7 \pi / 4\right) $$

Determine the eccentricity, identify the conic, and sketch its graph. $$ r=\frac{-4}{\cos \theta-1} $$

Sketch the curve that has the given set of parametric equations. $$ x=t^{3}+1, y=t^{2}-1,-2 \leq t \leq 2 $$

Find alternative polar coordinates that satisfy (a) \(r>0, \theta<0\) (b) \(r>0, \theta>2 \pi\) (c) \(r<0, \theta>0\) (d) \(r<0, \theta<0\) for each point with the given polar coordinates. $$ (2,3 \pi / 4) $$

Determine the eccentricity, identify the conic, and sketch its graph. $$ r=\frac{18}{3-6 \cos \theta} $$

Sketch the curve that has the given set of parametric equations. $$ x=3 \cos t, y=5 \sin t, 0 \leq t \leq 2 \pi $$

Find alternative polar coordinates that satisfy (a) \(r>0, \theta<0\) (b) \(r>0, \theta>2 \pi\) (c) \(r<0, \theta>0\) (d) \(r<0, \theta<0\) for each point with the given polar coordinates. $$ (5, \pi / 2) $$

Determine the eccentricity, identify the conic, and sketch its graph. $$ r=\frac{4 \csc \theta}{3 \csc \theta+2} $$

Sketch the curve that has the given set of parametric equations. $$ x=3+2 \sin t, y=4+\sin t,-\pi / 2 \leq t \leq \pi / 2 $$

Find alternative polar coordinates that satisfy (a) \(r>0, \theta<0\) (b) \(r>0, \theta>2 \pi\) (c) \(r<0, \theta>0\) (d) \(r<0, \theta<0\) for each point with the given polar coordinates. $$ (4, \pi / 3) $$

Determine the eccentricity, identify the conic, and sketch its graph. $$ r=\frac{6}{1-\cos \theta} $$

Sketch the curve that has the given set of parametric equations. $$ x=e^{t}, y=e^{3 t}, 0 \leq t \leq \ln 2 $$

Find alternative polar coordinates that satisfy (a) \(r>0, \theta<0\) (b) \(r>0, \theta>2 \pi\) (c) \(r<0, \theta>0\) (d) \(r<0, \theta<0\) for each point with the given polar coordinates. $$ (3, \pi / 4) $$

Determine the eccentricity, identify the conic, and sketch its graph. $$ r=\frac{2}{2+5 \cos \theta} $$

Eliminate the parameter from the given set of parametric equations and obtain a rectangular equation that has the same graph. $$ x=t^{2}, y=t^{4}+3 t^{2}-1 $$

Find alternative polar coordinates that satisfy (a) \(r>0, \theta<0\) (b) \(r>0, \theta>2 \pi\) (c) \(r<0, \theta>0\) (d) \(r<0, \theta<0\) for each point with the given polar coordinates. $$ (1, \pi / 6) $$

Eliminate the parameter from the given set of parametric equations and obtain a rectangular equation that has the same graph. $$ x=t^{3}+t+4, y=-2\left(t^{3}+t\right) $$

$$ r=2+4 \sin \theta $$

Find alternative polar coordinates that satisfy (a) \(r>0, \theta<0\) (b) \(r>0, \theta>2 \pi\) (c) \(r<0, \theta>0\) (d) \(r<0, \theta<0\) for each point with the given polar coordinates. $$ (3,7 \pi / 6) $$

Determine the eccentricity \(e\) of the given conic. Then convert the polar equation to a rectangular equation and verify that \(e=c / a\) . $$ r=\frac{10}{2-3 \cos \theta} $$

Eliminate the parameter from the given set of parametric equations and obtain a rectangular equation that has the same graph. $$ x=t^{3}+t+4, y=-2\left(t^{3}+t\right) $$

Find alternative polar coordinates that satisfy (a) \(r>0, \theta<0\) (b) \(r>0, \theta>2 \pi\) (c) \(r<0, \theta>0\) (d) \(r<0, \theta<0\) for each point with the given polar coordinates. $$ (9,3 \pi / 2) $$

Determine the eccentricity \(e\) of the given conic. Then convert the polar equation to a rectangular equation and verify that \(e=c / a\) . $$ r=\frac{12}{3-2 \cos \theta} $$

Eliminate the parameter from the given set of parametric equations and obtain a rectangular equation that has the same graph. $$ x=\cos 2 t, y=\sin t,-\pi / 2 \leq t \leq \pi / 2 $$

Find alternative polar coordinates that satisfy (a) \(r>0, \theta<0\) (b) \(r>0, \theta>2 \pi\) (c) \(r<0, \theta>0\) (d) \(r<0, \theta<0\) for each point with the given polar coordinates. $$ (5, \pi) $$

Determine the eccentricity \(e\) of the given conic. Then convert the polar equation to a rectangular equation and verify that \(e=c / a\) . $$ r=\frac{2 \sqrt{3}}{\sqrt{3}+\sin \theta} $$

Eliminate the parameter from the given set of parametric equations and obtain a rectangular equation that has the same graph. $$ x=e^{t}, y=\ln t, t>0 $$

Find the rectangular coordinates for each point with the given polar coordinates. $$ \left(\frac{1}{2}, 2 \pi / 3\right) $$

Convert the polar equation to a rectangular equation. Use the rectangular equation to verify that the focus of the conic is at the origin. $$ r=\frac{2}{1+\sin \theta} $$

Eliminate the parameter from the given set of parametric equations and obtain a rectangular equation that has the same graph. $$ x=t^{3}, y=3 \ln t, t>0 $$

Find the rectangular coordinates for each point with the given polar coordinates. $$ (-1,7 \pi / 4) $$

Convert the polar equation to a rectangular equation. Use the rectangular equation to verify that the focus of the conic is at the origin. $$ r=\frac{1}{1-\cos \theta} $$

Eliminate the parameter from the given set of parametric equations and obtain a rectangular equation that has the same graph. $$ x=\tan t, y=\sec t,-\pi / 2

Find the rectangular coordinates for each point with the given polar coordinates. $$ (-6,-\pi / 3) $$

Find a polar equation of the conic with focus at the origin that satisfies the given conditions. $$ e=1, \operatorname{directrix} x=3 $$