91Ó°ÊÓ

Finding the Vertex, Focus, and Directrix of a Parabola In Exercises \(29-42,\) find the vertex, focus, and directrix of the parabola. Then sketch the parabola. $$y^{2}-4 y-4 x=0$$

Finding the Standard Equation of a Parabola In Exercises \(47-56\) , find the standard form of the equation of the parabola with the given characteristics. $$(-1,2) ; \text { focus: }(-1,0)$$

Rectangular-to-Polar Conversion In Exercises \(43-60,\) a point in rectangular coordinates is given. Convert the point to polar coordinates. $$(-\sqrt{3},-\sqrt{3})$$

A lithotripter machine uses an elliptical reflector to break up kidney stones nonsurgically. A spark plug in the reflector generates energy waves at one focus of an ellipse. The reflector directs these waves toward the kidney stone, positioned at the other focus of the ellipse, with enough energy to break up the stone (see figure). The lengths of the major and minor axes of the ellipse are 280 millimeters, 160 millimeters, respectively. How far is the spark plug from the kidney stone?

In Exercises \(59-64,\) use a graphing utility to graph the polar equation. Find an interval for \(\theta\) for which the graph is traced only once. $$r^{2}=16 \sin 2 \theta$$

Satellite Dish The parabolic cross section of a satellite dish can be modeled by a portion of the graph of the equation $$x ^ { 2 } - 2 x y - 27 \sqrt { 2 } x + y ^ { 2 } + 9 \sqrt { 2 } y + 378 = 0$$ where all measurements are in feet. (a) Rotate the axes to eliminate the \(x y\) -term in the equation. Then write the equation in standard form. (b) A receiver is located at the focus of the cross section. Find the distance from the vertex of the cross section to the receiver.

In Exercises \(59-64,\) use a graphing utility to graph the polar equation. Find an interval for \(\theta\) for which the graph is traced only once. $$r^{2}=1 / \theta$$

A satellite in a 100 -mile-high circular orbit around Earth has a velocity of approximately \(17,500\) miles per hour. If this velocity is multiplied by \(\sqrt{2}\) , then the satellite will have the minimum velocity necessary to escape Earth's gravity and will follow a parabolic path with the center of Earth as the focus (a) Find a polar equation of the parabolic path of the satellite (assume the radius of Earth is 4000 miles. (b) Use a graphing utility to graph the equation you found in part ( a). (c) Find the distance between the surface of the Earth and the satellite when \(\theta=30^{\circ} .\) (d) Find the distance between the surface of Earth and the satellite when \(\theta=60^{\circ} .\)

The sound pickup pattern of a microphone is modeled by the polar equation \(r=5+5 \cos \theta\) where \(|r|\) measures how sensitive the microphone is to sounds coming from the angle \(\theta\) . (a) Sketch the graph of the model and identify the type of polar graph. (b) At what angle is the microphone most sensitive to sound?

The area of the lemniscate \(r^{2}=a^{2} \cos 2 \theta\) is \(a^{2}\) (a) Sketch the graph of \(r^{2}=16 \cos 2 \theta\) (b) Find the area of one loop of the graph from part (a).