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 3 x 2 + 4 x y = 4 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 = 3 , B = 4 and C = 0 . Compute the angle of rotation. tan 2 θ = 4 3 − 0 = 4 3 Sketch the graph for tan 2 θ = 4 3 as shown below in Figure 1. From Figure 1, it is observed that the opposite and adjacent sides of triangle is 4 and 3, respectively. Find the hypotenuse of a triangle. Hypotenuse = 3 2 + 4 2 = 9 + 16 = 25 = 5 Thus, the value cos 2 θ = 3 5 . The value of sin θ is computed as follows. sin θ = 1 − cos 2 θ 2 = 1 − 3 5 2 = 2 5 10 = 1 5 The value of cos θ is computed as follows. cos θ = 1 + cos 2 θ 2 = 1 + 3 5 2 = 8 5 2 = 4 5 = 2 5 Obtain the rotation equations. Substitute sin θ = 1 5 and cos θ = 2 5 in x = x ′ cos θ − y ′ sin θ . x = x ′ ( 2 5 ) − y ′ ( 1 5 ) = 2 x ′ − y ′ 5 Substitute sin θ = 1 5 and cos θ = 2 5 in y = x ′ sin θ + y ′ cos θ

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ISBN: 9780134437705

Author: Washington

Publisher: PEARSON

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Chapter 21.9, Problem 15E

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To transform: The equation 3x2+4xy=4 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 14E

Chapter 21.9, Problem 16E

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To transform: The equation 3 x 2 + 4 x y = 4 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 3x2+4xy=4 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