91Ó°ÊÓ

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc}\text{Conic} & \text{Eccentricity} & \text{Directrix} \\\ \text{Ellipse} & e=\frac{1}{2} & y=1 \end{array}$$

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

A point in rectangular coordinates is given. Convert the point to polar coordinates. $$(-4,-3)$$

The planets travel in elliptical orbits with the sun at one focus. Assume that the focus is at the pole, the major axis lies on the polar axis, and the length of the major axis is \(2 a\) (see figure). Show that the polar equation of the orbit is \(r=a\left(1-e^{2}\right) /(1-e \cos \theta),\) where \(e\) is the eccentricity.

A line segment through a focus of an ellipse with endpoints on the ellipse and perpendicular to the major axis is called a latus rectum of the ellipse. Therefore, an ellipse has two latera recta. Knowing the length of the latera recta is helpful in sketching an ellipse because this information yields other points on the ellipse (see figure). Show that the length of each latus rectum is \(2 b^{2} / a.\)

Roads are often designed with parabolic surfaces to allow rain to drain off. A particular road that is 32 feet wide is 0.4 foot higher in the center than it is on the sides (see figure). (a) Find an equation of the parabola with its vertex at the origin that models the road surface. (b) How far from the center of the road is the road surface 0.1 foot lower than in the middle?

The comet Hale-Bopp has an elliptical orbit with an eccentricity of \(e \approx 0.995 .\) The length of the major axis of the orbit is approximately 500 astronomical units. Find a polar equation for the orbit. How close does the comet come to the sun?

A simply supported beam is 64 feet long and has a load at the center (see figure). The deflection (bending) of the beam at its center is 1 inch. The shape of the deflected beam is parabolic. (a) Find an equation of the parabola with its vertex at the origin that models the shape of the beam. (b) How far from the center of the beam is the deflection equal to \(\frac{1}{2}\) inch?

Determine whether the statement is true or false. Justify your answer. The graph of \(r=10 \sin 5 \theta\) is a rose curve with five petals.

A circle and a parabola can have \(0,1,2,3,\) or 4 points of intersection. Sketch the circle \(x^{2}+y^{2}=4 .\) Discuss how this circle could intersect a parabola with an equation of the form \(y=x^{2}+C .\) Then find the values of \(C\) for each of the five cases described below. Use a graphing utility to verify your results. (a) No points of intersection (b) One point of intersection (c) Two points of intersection (d) Three points of intersection (e) Four points of intersection