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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
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The given hyperbola is defined by the equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). The foci of this hyperbola are located at the points \( F_1(-c, 0) \) and \( F_2(c, 0) \), where \( c = \sqrt{a^2 + b^2} \). Show more…
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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 $P F_{1}, P F_{2}$ 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 $\tan \alpha=\tan \beta .]$
Parametric Equations and Polar Coordinates
Conic Sections
Let $ P(x_1, y_1) $ 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 is the reflection property of the hyperbola. It shows that light aimed at a focus $ F_2 $ of a hyperbolic mirror is reflected toward the other focus $ F_1 $.)
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 $P F_{1}, P F_{2}$ and the hyperbola as shown in the figure. Prove that $\alpha=\beta .$ This is the reflection property of the hyperbola. It shows that light aimed at a focus $F_{2}$ of a hyperbolic mirror is reflected toward the other focus $F_{1 .} .$ .
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