?Reflection Properties of Conic Sections We saw the reflection property of parabolas in

Chapter 9, Problem 68

(choose chapter or problem)

Reflection Properties of Conic Sections We saw the reflection property of parabolas in Problem 22 of Problems Plus following Chapter 3. Here we investigate the reflection properties of ellipses and hyperbolas.

Let  \(P\left(x_{1}, y_{1}\right)\) be a point on the hyperbola \(x^{2} / a^{2}-y^{2} / b^{2}=1\) with foci \(F_1\)and \(F_2\) and let \(\alpha\) and \(\beta\) be the angles between the lines \(PF_1,PF_2\) and the hyperbola as shown in the figure. Prove that \(\alpha=\beta\). This shows that light aimed at a focus  \(F_2\) of a hyperbolic mirror is reflected toward the other focus  \(F_1\).

                

Equation Transcription:

Text Transcription:

P(x_1,y_1)

x^2/a^2- y^2/b^2= 1

F_1

F_2

Alpha

Beta

PF_1

PF_2

alpha=beta