1126 APPENDIX B Rotation of Axes Application of a Rotation Equation x y x9 y9 45° x2 + y2 + 2xy + 2√2x – 2√2y = 0 x92 = 2y9 Figure 2 EXAMPLE 1 Finding an Equation after a Rotation The equation of a curve is x2 + y2 + 2xy + 222x - 222y = 0. Find the resulting equation if the axes are rotated 45°. Graph the equation. SOLUTION If u = 45°, then sin u = 22 2 and cos u = 22 2 , and the rotation equations become x = 22 2 x′ - 22 2 y′ and y = 22 2 x′ + 22 2 y′. Substitute these values into the given equation. x2 + y2 + 2xy + 222x - 222y = 0 c 22 2 x′ - 22 2 y′d 2 + c 22 2 x′ + 22 2 y′d 2 + 2 c 22 2 x′ - 22 2 y′d c 2 2 x′ + 2 2 y′d + 222 c 22 2 x′ - 22 2 y′d - 222 c 22 2 x′ + 22 2 y′d = 0 1 2 x′2 - x′y′ + 1 2 y′2 + 1 2 x′2 + x′y′ + 1 2 y′2 + x′2 - y′2 + 2x′ - 2y′ - 2x′ - 2y′ = 0 Expand terms. 2x′2 - 4y′ = 0 Combine like terms. x′2 - 2y′ = 0 Divide by 2. x′2 = 2y′ Add 2y′. This is the equation of a parabola. See Figure 2. S Now Try Exercise 13. We have graphed equations written in the general form Ax2 + Cy2 + Dx + Ey + F = 0. To graph an equation that has an xy-term by hand, it is necessary to find an appropriate angle of rotation to eliminate the xy-term. The necessary angle of rotation can be determined by using the following result. The proof is quite lengthy and is not presented here. Angle of Rotation The xy-term is removed from the general equation Ax2 +Bxy +Cy2 +Dx +Ey +F =0 by a rotation of the axes through an angle u, 0° 6u 690°, where cot 2U = A −C B . To find the rotation equations, first find sin u and cos u. Example 2 illustrates a way to obtain sin u and cos u from cot 2u without first identifying angle u.
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