722 CHAPTER 10 Analytic Geometry Figure 51 (b) Rotation of axes y x x u a O y r P 5 (x, y) 5 (x9, y9) y9 x9 x9 y9 Rotating Axes Express the equation = xy 1 in terms of new ′ ′ x y -coordinates by rotating the axes through a ° 45 angle. Discuss the new equation. Solution EXAMPLE 2 Let θ = ° 45 in formulas (5). Then ( ) ( ) = ′ ° − ′ ° = ′ − ′ = ′ − ′ = ′ ° + ′ ° = ′ + ′ = ′ + ′ x x y x y x y y x y x y x y cos45 sin45 2 2 2 2 2 2 sin45 cos45 2 2 2 2 2 2 Substituting these expressions for x and y in = xy 1 gives ( ) ( ) ( ) ′ − ′ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⋅ ′ + ′ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ′ − ′ = ′ − ′ = x y x y x y x y 2 2 2 2 1 1 2 1 2 2 1 2 2 2 2 This is the equation of a hyperbola with center at ( ) 0, 0 and transverse axis along the ′x -axis. The vertices are at ( ) ± 2, 0 on the ′x -axis; the asymptotes are ′ = ′ y x and ′ = − ′ y x (which correspond to the original x - and y -axes). See Figure 52 for the graph of = xy 1. Figure 52 = xy 1 y x 458 1 21 22 2 2 1 21 22 y9 x9 2 1 , 02 2 12 , 02 Now look at Figure 51(b).There the point P has the coordinates ( ) x y , relative to the xy -plane, while the same point P has coordinates ( ) ′ ′ x y , relative to the ′ ′ x y -plane. We need relationships that enable us to express x and y in terms of ′ ′ x y , , and θ. As Figure 51(b) shows, r denotes the distance from the origin O to the point P, and α denotes the angle between the positive ′x -axis and the ray from O through P. Then, using the definitions of sine and cosine, we have α α ′ = ′ = x r y r cos sin (3) θ α θ α ( ) ( ) = + = + x r y r cos sin (4) Now θ α θ α θ α α θ α θ θ θ ( ) ( ) ( ) ( ) = + = − = ⋅ − ⋅ = ′ − ′ x r r r r x y cos cos cos sin sin cos cos sin sin cos sin Similarly, θ α θ α θ α θ θ ( ) ( ) = + = + = ′ + ′ y r r x y sin sin cos cos sin sin cos Use the Sum Formula for cosine. By equation (3) Use the Sum Formula for sine. By equation (3) THEOREM Rotation Formulas If the x - and y -axes are rotated through an angle θ, the coordinates ( ) x y , of a point P relative to the xy -plane and the coordinates ( ) ′ ′ x y , of the same point relative to the new ′x - and ′y -axes are related by the formulas θ θ θ θ = ′ − ′ = ′ + ′ x x y y x y cos sin sin cos (5)
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