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. $$ (2,3 \pi / 4) $$

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. $$ (5, \pi / 2) $$

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. $$ x=e^{t}, y=e^{3 t}, 0 \leq t \leq \ln 2 $$

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

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. $$ (4, \pi / 3) $$

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. $$ (3, \pi / 4) $$

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

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