Elliptical Billiard Table

Can you bounce the cue ball off a cushion and hit the second ball placed on the second spot?

Does this always work?

![](https://www.datocms-assets.com/177690/1760720759-alhazanproblem.png) If you think this doesn’t always work, what factors affect the success?

To draw an ellipse, loop a piece of string around two drawing pins. Use a pencil to pull the string taut and, while keeping it taut, move the pencil around the two the pins.

![](https://www.datocms-assets.com/177690/1760720758-ellipsedrawingb.png)

The pins are called focal points and the total distance between the ellipse and the two focal points is constant (it’s the length of the string). This is enough to define an ellipse. While drawing the ellipse, it can be shown that the angles between the ellipse and each focal point are equal - exactly like a ball bouncing off an elliptical billiard table.

So, any ball that passes through one focal point will rebound and pass through the other focal point. If the ball is allowed to continue, it will continue to pass through the two focal points, eventually aligning with the horizonal (major) axis.