91Ó°ÊÓ

Use a graphing utility to graph the polar equation and find the area of the given region.Between the loops of \(r=1+2 \cos \theta\)

The rectangular coordinates of a point are given. Plot the point and find two sets of polar coordinates for the point for \(0 \leq \theta<2 \pi\). $$ (\sqrt{3},-1) $$

Find the vertex, focus, and directrix of the parabola, and sketch its graph. $$ x^{2}+4 x+4 y-4=0 $$

Find an equation of the tangent line at each given point on the curve. $$ \begin{aligned} &x=2 \cot \theta \\ &y=2 \sin ^{2} \theta \end{aligned} $$

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. \(x=e^{t}, \quad y=e^{3 t}+1\)

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. \(x=e^{-t}, \quad y=e^{2 t}-1\)

Find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. $$ r=\frac{5}{5+3 \sin \theta} $$

Find the vertex, focus, and directrix of the parabola, and sketch its graph. $$ y^{2}+4 y+8 x-12=0 $$

The rectangular coordinates of a point are given. Plot the point and find two sets of polar coordinates for the point for \(0 \leq \theta<2 \pi\). $$ (3,-\sqrt{3}) $$

Find the vertex, focus, and directrix of the parabola. Then use a graphing utility to graph the parabola. $$ y^{2}+x+y=0 $$