SECTION 10.5 Rotation of Axes; General Form of a Conic 723 As Example 2 illustrates, a rotation of axes through an appropriate angle can transform a second-degree equation in x and y containing an xy -term into one in ′x and ′y in which no ′ ′ x y -term appears. In fact, a rotation of axes through an appropriate angle will transform any equation of the form of equation (1) into an equation in ′x and ′y without an ′ ′ x y -term. To find the formula for choosing an appropriate angle θ through which to rotate the axes, begin with equation (1), + + + + + = ≠ Ax Bxy Cy Dx Ey F B 0 0 2 2 Now rotate the x- and y -axes through an angle θ using the rotation formulas (5). θ θ θ θ θ θ θ θ θ θ θ θ ( ) ( )( ) ( ) ( ) ( ) ′ − ′ + ′ − ′ ′ + ′ + ′ + ′ + ′ − ′ + ′ + ′ + = A x y B x y x y C x y D x y E x y F cos sin cos sin sin cos sin cos cos sin sin cos 0 2 2 Expanding and collecting like terms gives A B C x B C A x y A B C y D E x D E y F cos sin cos sin cos sin 2 sin cos sin sin cos cos cos sin sin cos 0 2 2 2 2 2 2 2 2 θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) + + ′ + − + − ′ ′ + − + ′ + + ′ + − + ′ + = (6) In equation (6), the coefficient of ′ ′ x y is θ θ θ θ ( ) ( ) ( ) − + − B C A cos sin 2 sin cos 2 2 To eliminate the ′ ′ x y -term, select an angle θ so that this coefficient is 0. θ θ θ θ θ θ θ θ θ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) − + − = + − = = − = − ≠ B C A B C A B A C A C B B cos sin 2 sin cos 0 cos 2 sin 2 0 cos 2 sin 2 cot 2 0 2 2 Double-angle Formulas Equation (7) has an infinite number of solutions for θ. We follow the convention of choosing the acute angle θ that satisfies (7). There are two possibilities: • If θ ( ) ≥ cot 2 0, then θ ° < ≤ ° 0 2 90 , so θ ° < ≤ ° 0 45 . • If θ ( ) < cot 2 0, then θ ° < < ° 90 2 180 , so θ ° < < ° 45 90 . Each of these results in a counterclockwise rotation of the axes through an acute angle θ. * *Any rotation through an angle θ that satisfies A C B cot 2θ ( ) = − will eliminate the x y -term. ′ ′ However, the final form of the transformed equation may be different (but equivalent), depending on the angle chosen. CAUTION Be careful if you use a calculator to solve equation (7). • If θ ( ) = cot 2 0, then θ = ° 2 90 and θ = ° 45 . • If θ ( ) ≠ cot 2 0, first find θ ( ) cos 2 . Then use the inverse cosine function key(s) to obtain θ θ ° < < ° 2 , 0 2 180 . Finally, divide by 2 to obtain the correct acute angle θ. j THEOREM Transformation Angle To transform the equation + + + + + = ≠ Ax Bxy Cy Dx Ey F B 0 0 2 2 into an equation in ′x and ′y without an ′ ′ x y -term, rotate the axes through an angle θ that satisfies the equation θ ( ) = −A C B cot 2 (7)
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