Reflective Property of an Ellipse

Reflective property. The two lines from an ellipse's foci to any point $P$ on it make equal angles with the tangent at $P$ (Fig. 5.62) — so a wave (light, sound, or a shock wave) emitted from one focus and striking the ellipse anywhere is reflected to arrive exactly at the other focus (Fig. 5.63).

This is the mechanism behind a whispering gallery: a room with an elliptical ceiling lets a whisper at one focus be heard clearly at the other, however far apart, because every reflected sound path converges there. It is also the working principle of a lithotripter (Fig. 5.4, 5.63): a high-frequency (electromagnetic or ultrasound) shock wave, generated at one focus of an ellipsoidal reflector, is focused onto a kidney stone positioned at the other focus, pulverising it non-invasively — with a much shorter recovery time, and lower mortality risk, than conventional surgery. …