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 9 x 2 − 24 x y + 16 y 2 − 320 x − 240 y = 0 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 = 9 , B = − 24 and C = 16 . Compute the angle of rotation. tan 2 θ = − 24 9 − 16 = 24 7 Sketch the graph for tan 2 θ = 24 7 as shown below in Figure 1. From Figure 1, it is observed that the opposite and adjacent sides of triangle is 24 and 7, respectively. Find the hypotenuse of a triangle. Hypotenuse = 24 2 + 7 2 = 576 + 49 = 625 = 25 Thus, the value cos 2 θ = 7 25 . The value of sin θ is computed as follows. sin θ = 1 − cos 2 θ 2 = 1 − 7 25 2 = 18 50 = 9 25 = 3 5 The value of cos θ is computed as follows. cos θ = 1 + cos 2 θ 2 = 1 + 7 25 2 = 32 50 = 16 25 = 4 5 Obtain the rotation equations. Substitute sin θ = 3 5 and cos θ = 4 5 in x = x ′ cos θ − y ′ sin θ . x = x ′ ( 4 5 ) − y ′ ( 3 5 ) = 4 x ′ − 3 y ′ 5 Substitute sin θ = 3 5 and cos θ = 4 5 in y = x ′ sin θ + y ′ cos θ . y = x ′ ( 3 5 ) + y ′ ( 4 5 ) = 3 x ′ + 4 y ′ 5 Substitute x = 4 x ′ − 3 y ′ 5 and y = 3 x ′ + 4 y ′ 5 in 9 x 2 − 24 x y + 16 y 2 − 320 x − 240 y = 0

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Chapter 21.9, Problem 16E

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To transform: The equation 9x2−24xy+16y2−320x−240y=0 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 15E

Chapter 21.9, Problem 17E

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To transform: The equation 9 x 2 − 24 x y + 16 y 2 − 320 x − 240 y = 0 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 9x2−24xy+16y2−320x−240y=0 to a form without an xy-term by rotating the axis and then identify and sketch the curve.

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