Question

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 $ \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 on page 273 to show that $ \tan \alpha = \tan \beta $.]

Name: Problem 65, Chapter 10: Calculus: Early Transcendentals
Uploaded: 2019-12-30T21:39:08-08:00
Duration: 3 min 45 s
Channel: Clarissa Noh

   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 $ \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 on page 273 to show that $ \tan \alpha = \tan \beta $.]

Calculus: Early Transcendentals

James Stewart 8th Edition

Chapter 10, Problem 65

Instant Answer

Solved by Expert Clarissa Noh

12/30/2019

Step 1

From the figure, we can see that $m_2 = \frac{y_1}{x_1+c}$ and $m_3 = \frac{y_1}{x_1-c}$. Show more…

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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 $ \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 on page 273 to show that $ \tan \alpha = \tan \beta $.]