91Ó°ÊÓ

For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula Area \(=a \cdot b \cdot \pi\). \(\frac{(x+1)^{2}}{4}+\frac{(y-2)^{2}}{5}=1\)

For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula Area \(=a \cdot b \cdot \pi\). \(4 x^{2}-8 x+9 y^{2}-72 y+112=0\)

For the following exercises, express the equation for the hyperbola as two functions, with \(y\) as a function of \(x\). Express as simply as possible. Use a graphing calculator to sketch the graph of the two functions on the same axes. \(4 x^{2}-24 x-y^{2}-4 y+16=0\)

Given \(6 x^{2}+12 x y+k y^{2}+16 x+10 y+4=0\) find \(k\) for the graph to be an ellipse.

For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation. \(V(4,-3)\), Endpoints \(\left(5,-\frac{7}{2}\right)\), \(\left(3,-\frac{7}{2}\right)\)

For the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph. The hedge will follow the asymptotes \(y=x \quad\) and \(y=-x,\) and its closest distance to the center fountain is 5 yards.

The mirror in an automobile headlight has a parabolic crosssection with the light bulb at the focus. On a schematic, the equation of the parabola is given as \(x^{2}=4 y\). At what coordinates should you place the light bulb?

Find the equation of the ellipse that will just fit inside a box that is 8 units wide and 4 units high.

For the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph. The hedge will follow the asymptotes \(y=2 x\) and \(y=-2 x,\) and its closest distance to the center fountain is 6 yards.

For the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph. The hedge will follow the asymptotes \(y=\frac{1}{2} x\) and \(y=-\frac{1}{2} x,\) and its closest distance to the center fountain is 10 yards.