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Stitz-Zeager_College_Algebra_e-book

# 22 v w v w v w v w v w

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Unformatted text preview: since this produces an acute angle,4 θ = arctan 3 . To 4 4 3 3 ﬁnd the rotation equations, we need cos(θ) = cos arctan 4 and sin(θ) = sin arctan 4 . Using the techniques developed in Section 10.6 we get cos(θ) = 4 and sin(θ) = 3 . Our rotation 5 5 y y x x equations are x = x cos(θ) − y sin(θ) = 45 − 35 and y = x sin(θ) + y cos(θ) = 35 + 45 . As usual, we now substitute these quantities into 16x2 + 24xy + 9y 2 + 15x − 20y = 0 and simplify. As a ﬁrst step, the reader can verify x2 = 16(x )2 24x y 9(y )2 − + , 25 25 25 xy = 12(x )2 7x y 12(y )2 + − , 25 25 25 y2 = 9(x )2 24x y 16(y )2 + + 25 25 25 Once the dust settles, we get 25(x )2 − 25y = 0, or y = (x )2 , whose graph is a parabola opening along the positive y -axis with vertex (0, 0). We graph this equation below. y y y y x x θ = arctan θ= π 4 3 4 x x √ √ 5x2 + 26xy + 5y 2 − 16x 2 + 16y 2 − 104 = 0 16x2 + 24xy + 9y 2 + 15x − 20y = 0 4 As usual, there are inﬁnitely many solutions to tan(θ) = 3 . We choose the acut...
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