Chapter 10

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

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

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

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

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

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

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

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

(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. $$36 x^{2}+9 y^{2}+48 x-36 y+43=0$$

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Vertex or Vertices} \\\ \text{Parabola} &(5, \pi)\end{array}$$

Use a graphing utility to find one set of polar coordinates for the point given in rectangular coordinates. (There are many correct answers.) $$\left(\frac{5}{2}, \frac{4}{3}\right)$$

Use a graphing utility to graph the polar equation. Describe your viewing window. $$r=8 \sin \theta \cos ^{2} \theta$$

Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: (1,2),(3,2)\(;\) asymptotes: \(y=x, y=4-x\)

Find the eccentricity of the ellipse. $$\frac{x^{2}}{4}+\frac{y^{2}}{9}=1$$

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Vertex or Vertices} \\\ \text{Parabola} &\left(10, \frac{\pi}{2}\right)\end{array}$$

Use a graphing utility to find one set of polar coordinates for the point given in rectangular coordinates. (There are many correct answers.) $$\left(-\frac{7}{4},-\frac{3}{2}\right)$$

Use a graphing utility to graph the polar equation. Describe your viewing window. $$r=2 \cos (3 \theta-2)$$

Find a set of parametric equations to represent the graph of the given rectangular equation using the parameters (a) \(t=x\) and (b) \(t=2-x.\) $$y=4 x-9$$

Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: (3,0),(3,-6) asymptotes: \(y=x-6, y=-x\)

Find the eccentricity of the ellipse. $$\frac{x^{2}}{25}+\frac{y^{2}}{49}=1$$

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Vertex or Vertices} \\\ \text{Ellipse} &(2,0),(10, \pi)\end{array}$$

Convert the rectangular equation to polar form. Assume \(a<0\) $$x^{2}+y^{2}=9$$C

Use a graphing utility to graph the polar equation. Describe your viewing window. $$r=2 \csc \theta+6$$

Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: (0,2),(6,2) asymptotes: \(y=\frac{2}{3} x, y=4-\frac{2}{3} x\)

Find the eccentricity of the ellipse. $$x^{2}+9 y^{2}-10 x+36 y+52=0$$

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Vertex or Vertices} \\\ \text{Ellipse} &\left(2, \frac{\pi}{2}\right),\left(4, \frac{3 \pi}{2}\right)\end{array}$$

Convert the rectangular equation to polar form. Assume \(a<0\) $$x^{2}+y^{2}=16$$

Use a graphing utility to graph the polar equation. Describe your viewing window. $$r=4-\sec \theta$$

Find a set of parametric equations to represent the graph of the given rectangular equation using the parameters (a) \(t=x\) and (b) \(t=2-x.\) $$y=\frac{1}{x^{2}}$$

Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: (3,0),(3,4) asymptotes: \(y=\frac{2}{3} x, y=4-\frac{2}{3} x\)

Find the eccentricity of the ellipse. $$4 x^{2}+3 y^{2}-8 x+18 y+19=0$$

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

Find the standard form of the equation of the hyperbola with the given characteristics. Foci: (-1,3),(9,3) asymptotes: \(y=\frac{3}{4} x, y=6-\frac{3}{4} x\)

Find an equation of the ellipse with the given characteristics. Vertices: (±5,0)\(;\) eccentricity: \(\frac{3}{5}\)

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Vertex or Vertices} \\\ \text{Ellipse} &(20,0),(4, \pi)\end{array}$$

Convert the rectangular equation to polar form. Assume \(a<0\) $$y=-\sqrt{3} x$$

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

Find a set of parametric equations to represent the graph of the given rectangular equation using the parameters (a) \(t=x\) and (b) \(t=2-x.\) $$y=x^{3}-x^{2}$$

Find the standard form of the equation of the hyperbola with the given characteristics. Foci: (1,2),(1,6) asymptotes: \(y=2+2 x, y=6-2 x\)

Find an equation of the ellipse with the given characteristics. Vertices: (0,±8)\(;\) eccentricity: \(\frac{1}{2}\)

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Vertex or Vertices} \\\ \text{Hyperbola} &\left(1, \frac{3 \pi}{2}\right),\left(-9, \frac{\pi}{2}\right)\end{array}$$

Convert the rectangular equation to polar form. Assume \(a<0\) $$y=a$$

Use a graphing utility to graph the polar equation. Find an interval for \(\theta\) for which the graph is traced only once. $$r=3-4 \cos \theta$$

You and a friend live 4 miles apart (on the same "east-west" street) and are talking on the phone. You hear a clap of thunder from lightning in a storm, and 18 seconds later your friend hears the thunder. Find an equation that gives the possible places where the lightning could have occurred. (Assume that the coordinate system is measured in feet and that sound travels at 1100 feet per second.)

Find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Focus: (0,2)

Find an equation of the ellipse with the given characteristics. Foci: (1,1),(1,13)\(;\) eccentricity: \(\frac{2}{3}\)

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Vertex or Vertices} \\\ \text{Hyperbola} &(2,0),(-8, \pi)\end{array}$$