1127 APPENDIX B Rotation of Axes x y 0 7 25 24 2u Figure 3 4 0 4 3 3 x y sin u = 3 5 tan u = 3 4 cos u = 4 5 + = 1 y92 2 52x2 – 72xy + 73y2 = 200 x92 8 x9 y9 u Figure 4 EXAMPLE 2 Rotating and Graphing Remove the xy-term from 52x2 - 72xy + 73y2 = 200 by performing a suitable rotation. Then graph the equation. SOLUTION Here A = 52, B = -72, and C = 73. cot 2u = A - C B = 52 - 73 -72 = -21 -72 = 7 24 Substitute into the angle of rotation equation, and simplify. To find sin u and cos u, use these trigonometric identities. sin u = B1 - cos 2u 2 and cos u = B1 + cos 2u 2 Sketch a right triangle as in Figure 3, to see that cos 2u = 7 25 . (In the two quadrants for which we are concerned, cosine and cotangent have the same sign.) sin u = D1 - 7 25 2 = B9 25 = 3 5 and cos u = D1 + 7 25 2 = B16 25 = 4 5 Use these values for sin u and cos u to obtain the following. x = 4 5 x′ - 3 5 y′ and y = 3 5 x′ + 4 5 y′ Substitute these expressions for x and y into the original equation. 52 c 4 5 x′ - 3 5 y′d 2 - 72 c 4 5 x′ - 3 5 y′d c 3 5 x′ + 4 5 y′d + 73 c 3 5 x′ + 4 5 y′d 2 = 200 52 c 16 25 x′2 - 24 25 x′y′ + 9 25 y′2d - 72 c 12 25 x′2 + 7 25 x′y′ - 12 25 y′2d + 73 c 9 25 x′2 + 24 25 x′y′ + 16 25 y′2d = 200 25x′2 + 100y′2 = 200 Combine like terms. x′2 8 + y′2 2 = 1 Divide by 200. This is an equation of an ellipse having x′-intercepts A{222, 0B and y′-intercepts A0, {22 B . The graph is shown in Figure 4. To find u, use the following. sin u cos u = 3 5 4 5 = 3 5 , 4 5 = 3 5 # 5 4 = 3 4 = tan u Use a calculator to find tan-1 3 4 ≈36.87°. S Now Try Exercise 17. Equation of a Conic with an xy-Term If the general second-degree equation Ax2 +Bxy +Cy2 +Dx +Ey +F =0 has a graph, it will be one of the following: (a) a circle or an ellipse or a point if B2 - 4AC60; (b) a parabola or one line or two parallel lines if B2 - 4AC = 0; (c) a hyperbola or two intersecting lines if B2 - 4AC70; (d) a straight line if A = B = C = 0, and D≠0 or E≠0.
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