Explanation Result used: (1) To find a rotation through a positive acute angle θ , use the equations x = x ′ cos θ − y ′ sin θ and y = x ′ sin θ + y ′ cos θ . (2) The angle of rotation is computed by the equation tan 2 θ = B A − C ( A ≠ C ) . The rotation of exactly 45 ° if A = C . Calculation: The given equation 11 x 2 − 6 x y + 19 y 2 = 20 is in the form of A x 2 + B x y + C y 2 + D x + E y + F = 0 Here, it is observed that A = 11 , B = − 6 and C = 19 . Compute the angle of rotation. tan 2 θ = − 6 11 − 19 = 6 8 = 3 4 Sketch the graph for tan 2 θ = 3 4 as shown below in Figure 1. From Figure 1, it is observed that the opposite and adjacent sides of triangle is 3 and 4, respectively. Find the hypotenuse of a triangle. Hypotenuse = 3 2 + 4 2 = 9 + 16 = 25 = 5 Thus, the value cos 2 θ = 4 5 . The value of sin θ is computed as follows. sin θ = 1 − cos 2 θ 2 = 1 − 4 5 2 = 1 10 = 1 10 The value of cos θ is computed as follows. cos θ = 1 + cos 2 θ 2 = 1 + 4 5 2 = 9 10 = 3 10 Obtain the rotation equations. Substitute sin θ = 1 10 and cos θ = 3 10 in x = x ′ cos θ − y ′ sin θ . x = x ′ ( 3 10 ) − y ′ ( 1 10 ) = 3 x ′ − y ′ 10 Substitute sin θ = 1 10 and cos θ = 3 10 in y = x ′ sin θ + y ′ cos θ

Mylab Math With Pearson Etext -- 18 Week Standalone Access Card -- For Basic Technical Mathematics With Calculus

11th Edition

ISBN: 9780135902912

Author: Allyn J. Washington

Publisher: PEARSON

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Chapter 21.9, Problem 17E

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To transform: The equation 11x2−6xy+19y2=20 to a form without an xy-term by rotating the axis and then identify and sketch the curve.

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Chapter 21.9, Problem 16E

Chapter 21.9, Problem 18E

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Chapter 21 Solutions

Mylab Math With Pearson Etext -- 18 Week Standalone Access Card -- For Basic Technical Mathematics With Calculus

Ch. 21.1 - Find the distance between (−2, 6) and (5, −3). Ch. 21.1 - Prob. 2PE Ch. 21.1 - Prob. 1E Ch. 21.1 - Prob. 2E Ch. 21.1 - Prob. 5E Ch. 21.1 - In Exercises 5–14, find the distance between the...Ch. 21.1 - Prob. 7E Ch. 21.1 - Prob. 8E Ch. 21.1 - Prob. 9E Ch. 21.1 - Prob. 10E

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To transform: The equation 11 x 2 − 6 x y + 19 y 2 = 20 to a form without an xy -term by rotating the axis and then identify and sketch the curve.

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To transform: The equation 11x2−6xy+19y2=20 to a form without an xy-term by rotating the axis and then identify and sketch the curve.

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Chapter 21 Solutions