Program Explanation Given information: The general quadratic equation is A x 1 2 + B x 1 x 2 + C x 2 2 = k represents an ellipse centered at the origin if and only if A k > 0 and discriminant B 2 − 4 A C < 0 . Explanation: It is given that the general quadratic equation is A x 1 2 + B x 1 x 2 + C x 2 2 = k represents an ellipse centered at the origin if and only if A k > 0 and discriminant B 2 − 4 A C < 0 . The given equation is − 4 x 1 2 + 6 x 1 x 2 − 17 2 x 2 2 = k . Now compare the given equation with general quadratic equation to obtain the value of A , B , C . The values of A , B , C are written below. A = − 4 , B = 6 , C = − 17 2 It can be seen that the equation − 4 x 1 2 + 6 x 1 x 2 − 17 2 x 2 2 = k is similar to the general quadratic equation. Therefore, the equation − 4 x 1 2 + 6 x 1 x 2 − 17 2 x 2 2 = k is also represent an ellipse that is centered at the origin if and only if A k > 0 and discriminant B 2 − 4 A C < 0 . Now calculate the value of B 2 − 4 A C from the equation − 4 x 1 2 + 6 x 1 x 2 − 17 2 x 2 2 = k . B 2 − 4 A C = 6 2 − 4 ( − 4 ) ( − 17 2 ) = 36 − 136 = − 100 < 0 According to the given fact of analytic geometry the value of A k > 0 . ( − 4 ) k > 0 − 4 k > 0 The condition − 4 k > 0 is possible only when the value of k is also negative. Hence, it has been proved. Now prove the converse part of the statement

Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis

5th Edition

ISBN: 9780321816252

Author: C. Henry Edwards, David E. Penney, David Calvis

Publisher: PEARSON

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Chapter 5.3, Problem 36P

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Program Description: Purpose of the problem is to show that all degenerate solution curves of the system −4x12+6x1x2−172x22=k are elliptical if k<0 .

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Chapter 5.3, Problem 35P

Chapter 5.3, Problem 37P

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Chapter 5 Solutions

Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis

Ch. 5.1 - Let A=[2347] and B=[3451]. Find (a) 2A+3B; (b)...Ch. 5.1 - Prob. 2P Ch. 5.1 - Find AB and BA given A=[203415] and B=[137032].Ch. 5.1 - Prob. 4P Ch. 5.1 - Prob. 5P Ch. 5.1 - Prob. 6P Ch. 5.1 - Prob. 7P Ch. 5.1 - Prob. 8P Ch. 5.1 - Prob. 9P Ch. 5.1 - Prob. 10P

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Program Description: Purpose of the problem is to show that all degenerate solution curves of the system − 4 x 1 2 + 6 x 1 x 2 − 17 2 x 2 2 = k are elliptical if k < 0 .

Solution Summary: The author explains that all degenerate solution curves of the system are elliptical if k0. Compare the given equation with the general quadratic equation to obtain the value of A

Concept explainers

Program Description: Purpose of the problem is to show that all degenerate solution curves of the system −4x12+6x1x2−172x22=k are elliptical if k<0 .

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Chapter 5 Solutions

Program Description: Purpose of the problem is to show that all degenerate solution curves of the system − 4 x 1 2 + 6 x 1 x 2 − 17 2 x 2 2 = k are elliptical if k &lt; 0 .

Solution Summary: The author explains that all degenerate solution curves of the system are elliptical if k0. Compare the given equation with the general quadratic equation to obtain the value of A

Concept explainers

Program Description: Purpose of the problem is to show that all degenerate solution curves of the system −4x12+6x1x2−172x22=k are elliptical if k<0 .

Want to see the full answer?

Chapter 5 Solutions

Program Description: Purpose of the problem is to show that all degenerate solution curves of the system − 4 x 1 2 + 6 x 1 x 2 − 17 2 x 2 2 = k are elliptical if k < 0 .