91Ó°ÊÓ

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

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

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

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

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

Eliminate the parameter from the given set of parametric equations and obtain a rectangular equation that has the same graph. $$ x=4 \cos t, y=2 \sin t, 0 \leq t \leq 2 \pi $$

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