91Ó°ÊÓ

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 it yields other points on the curve (see figure). Show that the length of each latus rectum is \(2 b^{2} / a\)

Each cable of the Golden Gate Bridge is suspended (in the shape of a parabola) between two towers that are 1280 meters apart. The top of each tower is 152 meters above the roadway. The cables touch the roadway midway between the towers. (a) Draw a sketch of the bridge. Locate the origin of a rectangular coordinate system at the center of the roadway. Label the coordinates of the known points. (b) Write an equation that models the cables. (c) Complete the table by finding the height \(y\) of the suspension cables over the roadway at a distance of \(x\) meters from the center of the bridge. $$\begin{array}{|c|c|}\hline \text { Distance, } \boldsymbol{x} & \text { Height, } \boldsymbol{y} \\\\\hline 0 & \\\100 & \\\250 & \\\400 & \\\500 & \\\\\hline\end{array}$$

Sketch the graph of the ellipse, using latera recta. \(5 x^{2}+3 y^{2}=15\)

The receiver in a parabolic satellite dish is 4.5 feet from the vertex and is located at the focus (see figure). Write an equation for a cross section of the reflector. (Assume that the dish is directed upward and the vertex is at the origin.)

Consider the parametric equations \(x=8 \cos t\) and \(y=8 \sin t\) (a) Describe the curve represented by the parametric equations. (b) How does the curve represented by the parametric \(\quad\) equations \(\quad x=8 \cos t+3\) and \(y=8 \sin t+6\) compare with the curve described in part (a)? (c) How does the original curve change when cosine and sine are interchanged?

A simply supported beam is 12 meters long and has a load at the center (see figure). The deflection of the beam at its center is 2 centimeters. Assume that the shape of the deflected beam is parabolic. (a) Write an equation of the parabola. (Assume that the origin is at the center of the deflected beam.) (b) How far from the center of the beam is the deflection equal to 1 centimeter?

Find the equation of an ellipse such that for any point on the ellipse, the sum of the distances from the point (2,2) and (10,2) is 36 .

Water is flowing from a horizontal pipe 48 feet above the ground. The falling stream of water has the shape of a parabola whose vertex (0,48) is at the end of the pipe (see figure). The stream of water strikes the ground at the point \((10 \sqrt{3}, 0)\). Find the equation of the path taken by the water.