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This essential trains for: AMC-10, AMC-12, AIME.
If a ray is shot at an elliptical contour from one of the foci, then it will be reflected through the other focus.
Although this makes the ellipse a poor design for sound propagation, let us prove the statement above, as it may come in handy as a problem solving tool.
When a ray reflects on an obstacle, the angle of incidence is equal to the angle of reflection. For our problem, we would have to prove that if A and B are foci, and a ray is shot from A to P, then it will reflect through B.
This is equivalent to showing that from any point P on an ellipse the two segments connecting this point with the foci make equal angles with the tangent line at point P.
The definition of the ellipse gives us:
Extend the segment AP beyond P by a length PEequal to PB. Construct the angle bisector of the angle BPE and denote with M the point where it intersects the segment BE.
Then, we have:
Since the triangle BPE is isosceles the angle bisector at P is also a perpendicular bisector of the segment PE and, therefore:
Now consider an point chosen arbitrarily on the line PM and denote it with Q.
Because PM is a perpendicular bisector of BE, we have that , for any point Q on the line PM:
where the triangle inequality has been used. Also,
with equality happening when Q=P, which means that, for any point Q on the line PM and point P on the ellipse, the point Q is outside the ellipse for any position other than P itself. This means that the line PM intersects the ellipse at a single point and is, therefore, a tangent to the ellipse.