91Ó°ÊÓ

Show that an equation of a conic having its principal axis along the polar axis and its extension, a focus at the pole, and the corresponding directrix to the right of the focus is \(r=e d /(1+e \cos \theta)\).

For each of the parabolas in Exercises 1 through 8 , find the coordinates of the focus, an equation of the directrix, and the length of the latus rectum. Draw a sketch of the curve. $$ y^{2}=6 x $$

Remove the \(x y\) term from the given equation by a rotation of axes. Draw a sketch of the graph and show both sets of axes. $$ 24 x y-7 y^{2}+36=0 $$

Find the vertices, foci, directrices, eccentricity, and ends of the minor axis of the given ellipse. Draw a sketch of the curve and show the foci and the directrices. $$ 3 x^{2}+4 y^{2}=9 $$

Remove the \(x y\) term from the given equation by a rotation of axes. Draw a sketch of the graph and show both sets of axes. $$ 6 x^{2}+20 \sqrt{3} x y+26 y^{2}=324 $$

Find an equation of the parabola having the given properties. Vertex, \((0,0) ;\) opens upward; length of latus rectum \(=3\)

The ceiling in a hallway \(20 \mathrm{ft}\) wide is in the shape of a semiellipse and is \(18 \mathrm{ft}\) high in the center and \(12 \mathrm{ft}\) high at the side walls. Find the height of the ceiling \(4 \mathrm{ft}\) from either wall.

A parabolic arch has a height of \(20 \mathrm{ft}\) and a width of \(36 \mathrm{ft}\) at the base. If the vertex of the parabola is at the top of the arch, at what height above the base is it \(18 \mathrm{ft}\) wide?

The cable of a suspension bridge hangs in the form of a parabola when the load is uniformly distributed horizontally. The distance between two towers is \(1500 \mathrm{ft}\), the points of support of the cable on the towers are \(220 \mathrm{ft}\) above the roadway, and the lowest point on the cable is \(70 \mathrm{ft}\) above the roadway. Find the vertical distance to the cable from a point in the roadway \(150 \mathrm{ft}\) from the foot of a tower.

Assume that water issuing from the end of a horizontal pipe, \(25 \mathrm{ft}\) above the ground, describes a parabolic curve, the vertex of the parabola being at the end of the pipe. If, at a point \(8 \mathrm{ft}\) below the line of the pipe, the flow of water has curved outward \(10 \mathrm{ft}\) beyond a vertical line through the end of the pipe, how far beyond this vertical line will the water strike the ground?