Hyperbola: Proof of Reflective Property

Autor:Brian Sterr

Tema:Hipérbola

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?

It appears to be tangent
bisects
is the perpendicular bisector of

Can we prove it is tangent? Let

be any point on

where

and connect it to

and

because

is on the perpendicular bisector of

by the triangle inequality

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 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.

Hyperbola: Proof of Reflective Property

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