Explanation Given: A focal chord of a conic section is a line through focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Calculation: Consider an ellipse centered at origin as shown below in Figure 1. The points F 1 and F 2 are the Foci of the ellipse and the point P is the intersection of the ellipse and the latus rectum. Let the focus be ( ± c , 0 ) and vertices be ( ± a , 0 ) . Let the length of P F 1 be l . Thus, the length of the latus rectum is 2l . Let the distance P F 2 be L then l + L = 2 a (1) Note that the distance F 1 F 2 is 2c . Use pythagorus theorem and obtain that L 2 = ( 2 c ) 2 + l 2 (2) Combine equations (1) and (2) obtain the following. ( 2 a − l ) 2 = 4 c 2 + l 2 4 a 2 + l 2 − 4 a l = 4 c 2 + l 2 4 a 2 − 4 c 2 = 4 a l l = a − c 2 a For an ellipse c 2 = a 2 − b 2 substitute in above equation

MyLab Math with Pearson eText -- 24 Month Access -- for Calculus with Integrated Review

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ISBN: 9780135243435

Author: Bill Briggs

Publisher: Pearson Education (US)

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Chapter 12.4, Problem 93E

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To show: The length of the latus rectum of the ellipse centered at the origin is 2b2a=2b1−e2.

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Chapter 12.4, Problem 92E

Chapter 12.4, Problem 94E

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The length of the latus rectum of the ellipse centered at the origin is 2 b 2 a = 2 b 1 − e 2 .

Solution Summary: The author calculates the length of the latus rectum of an ellipse centered at the origin by using pythagorus theorem.

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To show: The length of the latus rectum of the ellipse centered at the origin is 2b2a=2b1−e2.

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