91Ó°ÊÓ

Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises \(11-26 .\) $$0 \leq \theta \leq \pi, \quad r=-1$$

Find a parametrization for the curve. the line segment with endpoints \((-1,3)\) and \((3,-2)\)

In Exercises \(17-24\) , find the eccentricity of the hyperbola. Then find and graph the hyperbola's foci and directrices. $$y^{2}-3 x^{2}=3$$

Find the lengths of the curves in Exercises \(21-28 .\) The spiral \(r=e^{\theta} / \sqrt{2}, \quad 0 \leq \theta \leq \pi\)

Find the area enclosed by the ellipse $$ x=a \cos t, \quad y=b \sin t, \quad 0 \leq t \leq 2 \pi $$

Exercises \(17-24\) give equations for ellipses. Put each equation in standard form. Then sketch the ellipse. Include the foci in your sketch. $$ 6 x^{2}+9 y^{2}=54 $$

In Exercises \(17-24\) , find the eccentricity of the hyperbola. Then find and graph the hyperbola's foci and directrices. $$8 y^{2}-2 x^{2}=16$$

Graph the limacons. Limacon ("lee-ma-sahn") is Old French for "snail." You will understand the name when you graph the limacons in Exercise 21. Equations for limacons have the form \(r=a \pm b \cos \theta\) or \(r=a \pm b \sin \theta .\) There are four basic shapes. Dimpled limacons a. \(r=\frac{3}{2}+\cos \theta \quad\) b. \(r=\frac{3}{2}-\sin \theta\)

Find a parametrization for the curve. the lower half of the parabola \(x-1=y^{2}\)

Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises \(11-26 .\) $$\pi / 4 \leq \theta \leq 3 \pi / 4, \quad 0 \leq r \leq 1$$