Here we investigate the reflection properties of ellipses and hyperbolas Let P(x 1 , y 1 ) be a point on the ellipse x 2 a 2 y 2 b 2 1 with foci F 1 and F 2 and let and be the angles between the lines PF 1 , PF 2 and the ellipse as shown in the figure Prove that This explains how whispering galleri...

Differentiating implicitly Thus the slope ...

Here we investigate the reflection properties of ellipses and hyperbolas. Let P(x 1 , y 1 )

Question:

Here we investigate the reflection properties of ellipses and hyperbolas.

Let P(x₁, y₁) be a point on the ellipse x²/a² + y²/b² = 1 with foci F₁ and F₂ and let α and β be the angles between the lines PF₁, PF₂ and the ellipse as shown in the figure. Prove that α = β. This explains how whispering galleries and lithotripsy work. Sound coming from one focus is reflected and passes through the other focus.