91Ó°ÊÓ

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. $$ x=t^{3}+t+4, y=-2\left(t^{3}+t\right) $$

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. $$ (3,7 \pi / 6) $$

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

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. $$ x=\cos 2 t, y=\sin t,-\pi / 2 \leq t \leq \pi / 2 $$

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. $$ (9,3 \pi / 2) $$

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

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. $$ (5, \pi) $$