91Ó°ÊÓ

Find the coordinates of the focus and the equation of the directrix for each parabola. Make a sketch showing the parabola, its focus, and its directrix. \(3 x^{2}-9 y=0\)

Sketch the graph of the given polar equation and verify its symmetry. $$ r=\frac{4}{1+\sin \theta} $$

Name the conic that has the given equation. Find its vertices and foci, and sketch its graph. $$ 9 x^{2}+9 y^{2}-225=0 $$

In Problems \(1-10\), sketch the graph of the given equation and find the area of the region bounded by it. $$ r^{2}=6 \cos 2 \theta $$

Name the conic (horizontal ellipse, vertical hyperbola, and so on ) corresponding to the given equation. \(x^{2}-4 y^{2}=4\)

In Problems 9-14, find the standard equation of each parabola from the given information. Assume that the vertex is at the origin. Focus is at (2,0)

In Problems 9-16, sketch the graph of the given equation, indicating vertices, foci, and asymptotes (if it is a hyperbola). \(\frac{x^{2}}{16}+\frac{y^{2}}{4}=1\)

Sketch the graph of the given polar equation and verify its symmetry. $$ r=3-3 \cos \theta(\text { cardioid }) $$

In each of Problems, a parametric representation of a curve is given. (a) Graph the curve. (b) Is the curve closed? Is it simple? (c) Obtain the Cartesian equation of the curve by eliminating the parameter (see Examples \(1-4\) ). $$ x=t^{3}-4 t, y=t^{2}-4 ;-3 \leq t \leq 3 $$

Name the conic that has the given equation. Find its vertices and foci, and sketch its graph. $$ r=\frac{5}{2+2 \sin \theta} $$