Program Explanation Given information: The general quadratic equation is A x 1 2 + B x 1 x 2 + C x 2 2 = k represents in general conic section rotated by the angle θ such that tan 2 θ = B A − C . 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 in general conic section rotated by the angle θ such that tan 2 θ = B A − C . 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 represents in general conic section rotated by the angle θ such that tan 2 θ = B A − C . Now calculate the value of tan 2 θ = B A − C from the equation − 4 x 1 2 + 6 x 1 x 2 − 17 2 x 2 2 = k . tan 2 θ = 6 − 4 + 17 2 tan 2 θ = 12 9 tan 2 θ = 4 3 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 37P

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Program Description: Purpose of the problem is to show that angle θ=arctan24 and all degenerate elliptical solution curves of the system −4x12+6x1x2−172x22=k are rotated by the angle θ=arctan24 .

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Chapter 5.3, Problem 36P

Chapter 5.3, Problem 38P

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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 angle θ = arctan 2 4 and all degenerate elliptical solution curves of the system − 4 x 1 2 + 6 x 1 x 2 − 17 2 x 2 2 = k are rotated by the angle θ = arctan 2 4 .

Solution Summary: The author explains that the general quadratic equation represents in general conic section rotated by the angle theta.

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Program Description: Purpose of the problem is to show that angle θ=arctan24 and all degenerate elliptical solution curves of the system −4x12+6x1x2−172x22=k are rotated by the angle θ=arctan24 .

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Chapter 5 Solutions