91Ó°ÊÓ

Where must the foci of an ellipse lie?

What can be said about the symmetry of the graph of an ellipse with center at the origin and foci along the \(y\) -axis?

For the following exercises, find the foci for the given ellipses. $$ \frac{(x+1)^{2}}{100}+\frac{(y-2)^{2}}{4}=1 $$

For the following exercises, find the area of the ellipse is given by the formula Area \(=a \cdot b \cdot \pi\) $$ \frac{(x+6)^{2}}{16}+\frac{(y-6)^{2}}{36}=1 $$

Find the equation of the ellipse that will just fit inside a box that is four times as wide as it is high. Express in terms of \(h_{1},\) the height.

A person in a whispering gallery standing at one focus of the ellipse can whisper and be heard by a person standing at the other focus because all the sound waves that reach the ceiling are reflected to the other person. If a whispering gallery has a length of 120 feet, and the foci are located 30 feet from the center, find the height of the ceiling at the center.

For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci. $$\frac{y^{2}}{9}-\frac{x^{2}}{25}=1$$

For the following exercises, assume an object enters our solar system and we want to graph its path on a coordinate system with the sun at the origin and the x-axis as the axis of symmetry for the object's path. Give the equation of the flight path of each object using the given information. The object enters along a path approximated by the line \(y=0.5 x+2\) and passes within 1 au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line \(y=-0.5 x-2 .\)

For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form. $$y^{2}=4-x^{2}$$

For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form. $$y=4 x^{2}$$