91Ó°ÊÓ

Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises \(11-26 .\) $$\theta=\pi / 2, \quad r \leq 0$$

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. Limacons with an inner loop a. \(r=\frac{1}{2}+\cos \theta \quad\) b. \(r=\frac{1}{2}+\sin \theta\)

Exercises \(17-24\) give equations for ellipses. Put each equation in standard form. Then sketch the ellipse. Include the foci in your sketch. $$ 3 x^{2}+2 y^{2}=6 $$

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

Find the lengths of the curves in Exercises \(21-28 .\) The spiral \(r=\theta^{2}, \quad 0 \leq \theta \leq \sqrt{5}\)

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 \((4,1)\)

Find the area under one arch of the cycloid $$ x=a(t-\sin t), \quad y=a(1-\cos t) $$

Exercises \(17-24\) give equations for ellipses. Put each equation in standard form. Then sketch the ellipse. Include the foci in your sketch. $$ 9 x^{2}+10 y^{2}=90 $$

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