Explanation Results used: The two curves intersect each other at right angles if the tangent at that point of intersection are perpendicular to each other. Two lines are perpendicular to each other if the product of there slopes is −1. The slope of a tangent line to a curve y = f ( x ) is d d x ( y ) . Calculation: Let the equation of the ellipse be x 2 a 2 + y 2 b 2 = 1 and equation of the hyperbola be x 2 A 2 − y 2 B 2 = 1 . Assume that both the curves has foci at ( ± c , 0 ) . For an ellipse note that b 2 = a 2 − c 2 and for a hyperbola note that B 2 = c 2 − A 2 . Therefore the equation of the ellipse is x 2 a 2 + y 2 a 2 − c 2 = 1 and the equation of a hyperbola is x 2 A 2 − y 2 c 2 − A 2 = 1 . Let the point of intersection be P ( x 0 , y 0 ) . Substitute ( x 0 , y 0 ) in x 2 a 2 + y 2 a 2 − c 2 = 1 and x 2 A 2 + y 2 c 2 − A 2 = 1 and subtract the equations to obtain the following. x 0 2 a 2 + y 0 2 a 2 − c 2 − x 0 2 A 2 + y 0 2 c 2 − A 2 = 0 x 0 2 ( 1 a 2 − 1 A 2 ) + y 0 2 ( 1 a 2 − c 2 + 1 c 2 − A 2 ) = 0 x 0 2 ( A 2 − a 2 a 2 A 2 ) + y 0 2 ( a 2 − A 2 ( a 2 − c 2 ) ( c 2 − A 2 ) ) = 0 x 0 2 ( A 2 − a 2 a 2 A 2 ) = − y 0 2 ( a 2 − A 2 ( a 2 − c 2 ) ( c 2 − A 2 ) ) Simplify further as follows. x 0 2 y 0 2 = ( A 2 − a 2 ( a 2 − c 2 ) ( c 2 − A 2 ) ) ( A 2 − a 2 a 2 A 2 ) x 0 2 y 0 2 = a 2 A 2 ( a 2 − c 2 ) ( c 2 − A 2 ) Therefore, x 0 2 y 0 2 = a 2 A 2 ( a 2 − c 2 ) ( c 2 − A 2 ) (1) The slope of the tangent line to the ellipse is given as follows

Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

3rd Edition

ISBN: 9780134996103

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Chapter 12.4, Problem 95E

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To show: An ellipse and a hyperbola with same foci are perpendicularto each other.

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Chapter 12.4, Problem 94E

Chapter 12.4, Problem 96E

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Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

Ch. 12.1 - Identify the graph generated by the parametric...Ch. 12.1 - Prob. 2QC Ch. 12.1 - Prob. 3QC Ch. 12.1 - Prob. 4QC Ch. 12.1 - Prob. 5QC Ch. 12.1 - Prob. 6QC Ch. 12.1 - Explain how a pair of parametric equations...Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E Ch. 12.1 - Prob. 4E

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An ellipse and a hyperbola with same foci are perpendicularto each other.

Solution Summary: The author illustrates how an ellipse and a hyperbola intersect at right angles if the tangent at that point of intersection are perpendicular to each other.

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To show: An ellipse and a hyperbola with same foci are perpendicularto each other.

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