91Ó°ÊÓ

(a) find the standard form of the equation of the ellipse, (b) find the center, vertices, foci, and eccentricity of the ellipse, and (c) sketch the ellipse. Use a graphing utility to verify your graph. $$12 x^{2}+20 y^{2}-12 x+40 y-37=0$$

Find the standard form of the equation of the hyperbola with the given characteristics. Foci: (0,±8)\(;\) asymptotes: \(y=\pm 4 x\)

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Vertex or Vertices} \\\ \text{Parabola} &\left(1,-\frac{\pi}{2}\right)\end{array}$$

Use the results of Exercises 37-40 to find a set of parametric equations for the line or conic. Ellipse: vertices: \((±5,0)\); foci: \((±4,0)\)

Use a graphing utility to graph the polar equation. Describe your viewing window. $$r^{2}=\sin 2 \theta$$

Use a graphing utility to find one set of polar coordinates for the point given in rectangular coordinates. (There are many correct answers.) $$(-\sqrt{3}, 2)$$

Use the results of Exercises 37-40 to find a set of parametric equations for the line or conic. Hyperbola: vertices: \((±2,0)\); foci: \((±3,0)\)

Use a graphing utility to graph the polar equation. Describe your viewing window. $$r^{2}=4 \cos 3 \theta$$

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Vertex or Vertices} \\\ \text{Parabola} &(8,0)\end{array}$$

Find the standard form of the equation of the hyperbola with the given characteristics. Foci: (±10,0)\(;\) asymptotes: \(y=\pm \frac{3}{4} x\)