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Hyperbola: Proof of Reflective Property

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?
  1. It appears to be tangent
  2.  bisects 
  3.  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 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 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.