Explanation Procedure used: The standard equation of an ellipse is, A ( x − h ) 2 + B ( y − k ) 2 = C Here, A , B and C are nonzero real numbers of the same sign and ( h , k ) center point of the ellipse. Substitute h for x in equation to find the endpoints of major axis. Substitute k for y in equation to find the endpoints of minor axis. Calculation: Consider the given equation. 3 x 2 − 12 x + 5 y 2 + 10 y − 10 = 0 …… ( 1 ) Complete the square of the above equation as follows, ( 3 x 2 − 12 x ) + ( 5 y 2 + 10 y ) = 10 3 ( x 2 − 4 x ) + 5 ( y 2 + 2 y ) = 10 3 ( x 2 − 4 x + 4 ) + 5 ( y 2 + 2 y + 1 ) = 10 + 12 + 5 3 ( x − 2 ) 2 + 5 ( y + 1 ) 2 = 27 Express the above equation in the standard form as follows. 3 ( x − 2 ) 2 + 5 ( y − ( − 1 ) ) 2 = 27 …… ( 2 ) Compare the equation ( 2 ) with A ( x − h ) 2 + B ( y − k ) 2 = C . The value of h is 2 , k is − 1 . Thus, the center point of the ellipse is ( h , k ) = ( 2 , − 1 ) . Substitute 2 for x in equation ( 2 ) . 3 ( ( 2 ) − 2 ) 2 + 5 ( y − ( − 1 ) ) 2 = 27 0 + 5 ( y + 1 ) 2 = 27 ( y + 1 ) 2 = 27 5 Solve further for y , y + 1 = ± 27 5 y = − 1 ± 27 5 Solve for − 1 + 27 5 , y = − 1 + 27 5 Solve for − 1 − 27 5 , y = − 1 − 27 5 The intersecting points of the ellipse on x = 2 axis are ( 2 , − 1 − 27 5 ) , ( 2 , − 1 + 27 5 )

10th Edition

ISBN: 9781305191495

Author: Jerome E. Kaufmann; Karen L. Schwitters

Publisher: Cengage Learning US

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Chapter 8.CT, Problem 24CT

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The equation 3x2−12x+5y2+10y−10=0.

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Chapter 8.CT, Problem 23CT

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Intermediate Algebra

Ch. 8.1 - For Problems 110, answer true or false. The graph...Ch. 8.1 - Prob. 2CQ Ch. 8.1 - Prob. 3CQ Ch. 8.1 - Prob. 4CQ Ch. 8.1 - Prob. 5CQ Ch. 8.1 - Prob. 6CQ Ch. 8.1 - Prob. 7CQ Ch. 8.1 - Prob. 8CQ Ch. 8.1 - Prob. 9CQ Ch. 8.1 - Prob. 10CQ

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