SECTION 10.5 Rotation of Axes; General Form of a Conic 725 Analyzing an Equation Using a Rotation of Axes Analyze the equation − + + + = x xy y x 4 4 5 5 5 0. 2 2 Solution EXAMPLE 4 Letting = = − A B 4, 4, and = C 1 in equation (7), the acute angle θ to rotate the axes satisfies θ ( ) = − = − = − A C B cot 2 3 4 3 4 To use the rotation formulas (5), we need to know the values of θ sin and θ cos . Because the angle θ is acute, we know that θ > sin 0 and θ > cos 0. Use the Halfangle Formulas in the form θ θ θ θ ( ) ( ) = − = + sin 1 cos 2 2 cos 1 cos 2 2 See Figure 55. Because θ ( ) = − cot 2 3 4 , then θ ° < < ° 90 2 180 , so cos 2 3 5 . θ ( ) = − Then θ θ θ θ ( ) ( ) ( ) ( ) = − = − − = = = = + = + − = = = sin 1 cos 2 2 1 3 5 2 4 5 2 5 2 5 5 cos 1 cos 2 2 1 3 5 2 1 5 1 5 5 5 With these values, the rotation formulas (5) are ( ) ( ) = ′ − ′ = ′ − ′ = ′ + ′ = ′ + ′ x x y x y y x y x y 5 5 2 5 5 5 5 2 2 5 5 5 5 5 5 2 Substituting these values in the original equation and simplifying gives x xy y x x y x y x y x y x y 4 4 5 5 5 0 4 5 5 2 4 5 5 2 5 5 2 5 5 2 5 5 5 5 2 5 2 2 2 2 ( ) ( ) ( ) ( ) ( ) − + + + = ′ − ′ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ − ′ − ′ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ′ + ′ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ + ′ + ′ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ + ′ − ′ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ =− Multiply both sides by 5 and expand to obtain x x y y x x y y x x y y x y y y x y y x y y x y x 4 4 4 4 2 3 2 4 4 25 2 25 25 50 25 25 2 1 2 1 1 2 2 2 2 2 2 2 2 2 2 ( ) ( ) ( ) ( ) ′ − ′ ′ + ′ − ′ − ′ ′ − ′ + ′+ ′′+′+ ′− ′=− ′ − ′ + ′ =− ′ − ′ + ′ =− ′ − ′ + =− ′ ′ − =− ′ This is the equation of a parabola with vertex at ( ) 0, 1 in the ′ ′ x y -plane. The axis of symmetry is parallel to the ′x -axis. Use a calculator to solve θ = sin 2 5 5 , and find that θ ≈ ° 63.4 . See Figure 56 for the graph. Figure 56 ( ) ′ − =− ′ y x 1 2 63.48 y x x9 (0, 1) y9 Figure 55 θ ( ) = = − x r cos 2 3 5 4 y x (23, 4) 23 2u r 5 5 Combine like terms. Divide both sides by 25. Complete the square in ′y . Now Work PROBLEM 37
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