# Hyperbola: Proof of Reflective Property

- Author:
- Brian Sterr

- Topic:
- Hyperbola

Let be a point on the hyperbola
Draw in the focal radii.
Let be a point on such that .
Notice
Let be the midpoint of Draw line . What can you conclude about it?
be any point on where and connect it to , and .
because is on the perpendicular bisector of
by the triangle inequality
So is not on the hyperbola! Which means is the only point on , which is also on the hyperbola. In other words, is tangent to the hyperbola at .
Now extend the focal radii. We can see that all four angles formed are congruent to one another.
Since the angles formed with the tangent are congruent, we can see that a ray following the path starting

- It appears to be tangent
bisects is the perpendicular bisector of

**from**one focus will reflect off of the hyperbola directly**away**from the other focus. Similarly, a ray directed**towards**one focus will reflect off of the hyperbola**towards**the other focus.