?Reflection Properties of Conic Sections We saw the reflection property of parabolas in
Chapter 9, Problem 67(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 ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\) with foci \(F_1\)and \(F_2\) and let \(\alpha\) and \(\beta\) be the angles between the lines \(PF_1,PF_2\) and the ellipse as shown in the figure. Prove that \(\alpha=\beta\). This explains how whispering galleries and lithotripsy work. Sound coming from one focus is reflected and passes through the other focus. [Hint: Use the formula in Problem 21 in Problems Plus following Chapter 3 to show that \(\tan\alpha=\tan\beta\).]
Equation Transcription:
Text Transcription:
P(x sub 1,y sub 1)
x2/a2+ y2/b2= 1
F_1
F_2
Alpha
Beta
PF_1
PF_2
alpha=beta
Tan alpha = tan beta
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