724 CHAPTER 10 Analytic Geometry 3 Analyze an Equation Using a Rotation of Axes For the remainder of this section, the direction “Analyze the equation” means to transform the given equation so that it contains no xy-term and to graph the equation. Figure 53 ′ + ′ = x y 4 20 1 2 2 608 y x (2, 0) (22, 0) 1 0, 22 2 10, 2 2 y9 x9 5 5 Figure 54 3 23 24.8 4.8 Y2 Y1 Analyzing an Equation Using a Rotation of Axes Analyze the equation x xy y 3 2 10 0. 2 2 + + − = Solution EXAMPLE 3 Since an xy-term is present, rotate the axes. Using = = A B 1, 3, and = C 2 in equation (7), the acute angle θ of rotation satisfies the equation θ θ ( ) = − = − = − ° < < ° A C B cot 2 1 3 3 3 0 2 180 Since θ ( ) = − cot 2 3 3 , this means θ = ° 2 120 , so θ = ° 60 . Using formulas (5), x x y x y x y y x y x y x y cos60 sin60 1 2 3 2 1 2 3 sin60 cos60 3 2 1 2 1 2 3 ( ) ( ) = ′ ° − ′ ° = ′ − ′ = ′ − ′ = ′ ° + ′ ° = ′ + ′ = ′ + ′ Substituting these values into the original equation and simplifying gives x xy y x y x y x y x y 3 2 10 0 1 4 3 3 1 2 3 1 2 3 2 1 4 3 10 2 2 2 2 ( ) ( ) ( ) ( ) + + − = ′ − ′ + ⋅ ′ − ′ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⋅ ′ + ′ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ + ⋅ ′ + ′ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = Multiply both sides by 4 and expand to obtain x x y y x x y y x x y y x y x y 2 3 3 3 3 2 3 2 3 2 3 40 10 2 40 4 20 1 2 2 2 2 2 2 2 2 2 2 ( ) ( ) ′ − ′ ′ + ′ + ′ − ′ ′ − ′ + ′ + ′ ′ + ′ = ′ + ′ = ′ + ′ = This is the equation of an ellipse with center at ( ) 0, 0 and major axis along the ′y -axis. The vertices are at ( ) ± 0, 2 5 on the ′y -axis. See Figure 53 for the graph. To graph the equation + + − = x xy y 3 2 10 0 2 2 using some graphing utilities, such as the TI-84 Plus CE graphing calculator, we need to solve the equation for y. Rearranging the terms, observe that the equation is quadratic in the variable ( ) + + − = y y xy x : 2 3 10 0. 2 2 Solve the equation for y using the quadratic formula with = = a b x 2, 3 , and = − c x 10. 2 ( ) ( ) ( ) ( ) = − + − − = − + − + Y x x x x x 3 3 4 2 10 2 2 3 5 80 4 1 2 2 2 and ( ) ( ) ( ) ( ) = − − − − = − − − + Y x x x x x 3 3 4 2 10 2 2 3 5 80 4 2 2 2 2 Figure 54 shows the graph of Y1 and Y2 on a TI-84 Plus CE. Now Work PROBLEMS 21 AND 31 In Example 3, the acute angle θ of rotation was easy to find because of the numbers used in the given equation. In general, the equation θ ( ) = −A C B cot 2 does not have such a “nice” solution. As the next example shows, we can still find the appropriate rotation formulas without a calculator approximation by using Halfangle Formulas.
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