Applications of Ellipses

When a ray of light or sound emanating from one focus of an ellipse bounces off the ellipse, it passes through the other focus.

This reflecting property is responsible for “whispering galleries.” In a whispering gallery, a person whispering at a certain point in the room, though barely audible close by, can be heard clearly at another point across the room.

A lithotripter is a machine used to crush kidney stones using shock waves. The patient is placed in an elliptical tub with the kidney stone at one focus of the ellipse. A beam is projected from the other focus to the tub so that it reflects to hit the kidney stone.

Example

If a lithotripter is based on the ellipse $\frac{x^2}{36}+\frac{y^2}{27}=1$, determine how many units both the kidney stone and the source of the beam must be placed from the center of the ellipse.

The kidney stone and the source of the beam must be placed at the foci, (c, 0) and (− c, 0).

Here $a^2=36\:$and $b^2=27$, so $c=\sqrt[]{a^2-b^2}=\sqrt[]{36-27}=\sqrt[]{9}=3$.

The foci are (3,0) and (−3,0). The kidney stone and the source must be placed on a line 3 units from the center.

Please turn to the next page and complete Homework 10.2.