91Ó°ÊÓ

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square (see Examples 3-5). $$ 4 x^{2}-4 y^{2}-2 x+2 y+1=0 $$

Sketch the limaçon \(r=3-4 \sin \theta,\) and find the area of the region inside its small loop.

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

Find the standard equation of each parabola from the given information. Assume that the vertex is at the origin. Directrix is \(y-2=0\)

Find the Cartesian equation of the conic with the given properties. Vertices (±4,0) and eccentricity \(\frac{1}{2}\)

Sketch the graph of the given Cartesian equation, and then find the polar equation for it. $$ x-3 y+2=0 $$

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 \(1-4\) ). $$ x=2 \sqrt{t-2}, y=3 \sqrt{4-t} ; 2 \leq t \leq 4 $$

Sketch the graph of the given polar equation and verify its symmetry. $$ r=1-\sin \theta(\text { cardioid }) $$

Name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square (see Examples 3-5). $$ 4 x^{2}-4 y^{2}+8 x+12 y-5=0 $$

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 \(1-4\) ). $$ x=3 \sqrt{t-3}, y=2 \sqrt{4-t} ; 3 \leq t \leq 4 $$