Explanation Given: An ellipse is a set of points in a plane whose distance from two fixed points has a constant sum 2 a and the two fixed points are on the x -axis and are equidistant from the origin. Calculation: Let the two fixed points be F 1 ( c , 0 ) and F 2 ( − c , 0 ) . Consider a fixed point P on the y -axis having coordinates ( 0 , b ) such that P is equidistant from bot F 1 and F 2 and the distance is a . Note that the two points are on the x -axis and the point P is on the y -axis. Therefore, by pythagorus theorem b 2 + c 2 = a 2 . Consider a arbitrary point R ( x , y ) on the ellipse and without loss of generality let x , y > 0 . Diagramatically the condition is as shown below in Figure 1. From the Figure 1 observe that the distance F 2 R = ( x + c ) 2 + y 2 and the distance F 1 R = ( x − c ) 2 + y 2 . By the given defintion of the ellipse F 2 R + F 1 R = 2 a . Substitute F 2 R = ( x + c ) 2 + y 2 and F 1 R = ( x − c ) 2 + y 2 in F 2 R + F 1 R = 2 a and solve further. F 2 R + F 1 R = 2 a ( x + c ) 2 + y 2 + ( x − c ) 2 + y 2 = 2 a ( x + c ) 2 + y 2 = 2 a − ( x − c ) 2 + y 2 Take square on both sides and simplify. ( ( x + c ) 2 + y 2 ) 2 = ( 2 a − ( x − c ) 2 + y 2 ) 2 ( x + c ) 2 + y 2 = 4 a 2 + ( x − c ) 2 + y 2 − 4 a ( x − c ) 2 + y 2 4 a 2 + ( x − c ) 2 + y 2 − ( x + c ) 2 − y 2 = 4 a ( x − c ) 2 + y 2 4 a 2 + x 2 + c 2 − 2 x c − x 2 − c 2 − 2 x c = 4 a ( x − c ) 2 + y 2 Separate the square root term and rewrite the equation as follows

Calculus, Single Variable: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134766850

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

Publisher: PEARSON

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Vertices Of An Ellipse

In the study of the conic section of the geometry, An ellipse is a plane curve surrounding two focal points, or foci where the set of all locus of points in the plane has their sum of the distances to the two foci is constant. On the ellipse curve, it is…

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Chapter 12.4, Problem 85E

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To derive: The equation of the ellipse.

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Chapter 12.4, Problem 84E

Chapter 12.4, Problem 86E

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Calculus, Single Variable: Early Transcendentals (3rd Edition)

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To derive: The equation of the ellipse.

Solution Summary: The author explains the equation of the ellipse by pythagorus theorem.

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To derive: The equation of the ellipse.

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