Explanation Given: The given equation of the ellipse is, x 2 − 4 x + 9 y 2 + 54 y + 76 = 0 Approach: The standard equation of ellipse whose center is not at origin is, A ( x − h ) 2 + B ( y − k ) 2 = C Here, A , B and C are nonzero real numbers and center is at ( h , k ) . The formula to calculate the distance of a line segment with the given endpoints ( x 1 , y 1 ) and ( x 2 , y 2 ) is, d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 Here, d is the length of the line segment. Calculation: Consider the given equation of ellipse. x 2 − 4 x + 9 y 2 + 54 y + 76 = 0 Express the above equation in the standard form by completing the square. x 2 − 4 x + 9 y 2 + 54 y + 76 = 0 ( x 2 − 4 x + 2 2 ) + 9 ( y 2 + 6 y + 3 2 ) = − 76 + 2 2 + 9 ( 3 2 ) ( x 2 − 4 x + 4 ) + 9 ( y 2 + 6 y + 9 ) = − 76 + 4 + 81 ( x − 2 ) 2 + 9 ( y + 3 ) 2 = 9 ... ( 1 ) Compare the equation ( 1 ) with the standard equation A ( x − h ) 2 + B ( y − k ) 2 = C . The value of h is 2 and k is − 3 . Thus, the center of the ellipse is ( h , k ) = ( 2 , − 3 ) . To find the horizontal endpoints, substitute − 3 for y in equation ( 1 ) . ( x − 2 ) 2 + 9 ( ( − 3 ) + 3 ) 2 = 9 ( x − 2 ) 2 = 9 x − 2 = ± 3 x = 2 ± 3 Solve for the two values of x . x = 2 + 3 = 5 And, x = 2 − 3 = − 1 Thus, the end points are ( − 1 , − 3 ) and ( 5 , − 3 ) . To find the vertical endpoints, substitute 2 for x in equation ( 1 )

10th Edition

ISBN: 9781305191495

Author: Jerome E. Kaufmann; Karen L. Schwitters

Publisher: Cengage Learning US

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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 8.CR, Problem 21CR

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The length of the major axis and the length of the minor axis for the ellipse of given equation.

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Chapter 8.CR, Problem 20CR

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The length of the major axis and the length of the minor axis for the ellipse of given equation.

Solution Summary: The author calculates the length of the major and minor axis for the ellipse of a given equation by completing the square.

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To find: The length of the major axis and the length of the minor axis for the ellipse of given equation.

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Chapter 8 Solutions

To find:

The length of the major axis and the length of the minor axis for the ellipse of given equation.