Explanation Calculation: Consider the provided information as, 7 x 2 − 6 3 x y + 13 y 2 − 64 = 0 Rotate the axes to eliminate the x y -term in the equation 7 x 2 − 6 3 x y + 13 y 2 − 64 = 0 . The general second-degree equation is, A x 2 + B x y + C y 2 + D x + E y + F = 0 If B ≠ 0 , then the above equation can be written as, A ′ x ′ 2 + C ′ y ′ 2 + D ′ x ′ + E ′ y ′ + F ′ = 0 Where, cot 2 θ = A − C B Compare the provided equation with the second-degree equation as, A = 7 , B = − 6 3 and C = 13 . Rotate the coordinate axes through an angle θ . Now, to evaluate the value of θ , use the formula cot 2 θ = A − C B and substitute the values of A , B and C in it. cos 2 θ = A − C B = 17 − 3 − 6 3 = 1 3 For cot 2 θ = 0 , the value of θ is evaluated as: 2 θ = π 3 θ = π 6 Then, obtain the equation in the x ′ y ′ − system by substitution as, x = x ′ cos θ − y ′ sin θ y = x ′ sin θ + y ′ cos θ Substitute the value in the above equation as, y = x ′ sin θ + y ′ cos θ y = x ′ sin π 6 + y ′ cos π 6 y = x ′ 1 2 − y ′ 3 2 y = x ′ + 3 y ′ 2 And, x = x ′ cos θ − y ′ sin θ x = x ′ cos π 6 − y ′ sin π 6 x = x ′ 3 2 − y ′ 1 2 x = 3 x ′ − y ′ 2 Substitute this value in the provided equation as, 7 x 2 − 6 3 x y + 13 y 2 − 64 = 0 7 3

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ISBN: 9781337278461

Author: Ron Larson

Publisher: Cengage Learning

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Chapter 6.5, Problem 20E

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To calculate: The given equation 7x2−63xy+13y2−64=0 in standard form after rotating the axes to remove the xy-term. Also, sketch its graph.

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Chapter 6.5, Problem 19E

Chapter 6.5, Problem 21E

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The given equation 7 x 2 − 6 3 x y + 13 y 2 − 64 = 0 in standard form after rotating the axes to remove the x y -term . Also, sketch its graph.

Solution Summary: The author calculates the given equation 7x2-6sqrt3Xy+13y2-64=0 in standard form after rotating the axes to remove the

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To calculate: The given equation 7x2−63xy+13y2−64=0 in standard form after rotating the axes to remove the xy-term. Also, sketch its graph.

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Chapter 6 Solutions