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Stitz-Zeager_College_Algebra_e-book

# The dot product of v and w is given by v w v1 v2 w1

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Unformatted text preview: g equations. √ √ 1. 5x2 + 26xy + 5y 2 − 16x 2 + 16y 2 − 104 = 0 2. 16x2 + 24xy + 9y 2 + 15x − 20y = 0 Solution. √ √ 1. Since the equation 5x2 + 26xy + 5y 2 − 16x 2 + 16y 2 − 104 = 0 is already given to us in the form required by Theorem 11.10, we identify A = 5, B = 26 and C = 5 so that − cot(2θ) = A−C = 5265 = 0. This means cot(2θ) = 0 which gives θ = π + π k for integers k . B 4 2 We choose θ = π so that our rotation equations are x = 4 The reader should verify that 3 xy = (x )2 (y )2 − , 2 2 √ 2 2 − y √ 2 2 and y = x √ 2 2 + y √ 2 2 . (x )2 (y )2 +xy + 2 2 √ √ Making the other substitutions, we get that 5x2 + 26xy + 5y 2 − 16x 2 + 16y 2 − 104 = 0 2 2 reduces to 18(x )2 − 8(y )2 + 32y − 104 = 0, or (x4) − (y −2) = 1. The latter is the equation 9 of a hyperbola centered at the x y -coordinates (0, 2) opening in the x direction with vertices 3 (±2, 2) (in x y -coordinates) and asymptotes y = ± 2 x + 2. We graph it below. x2 = (x )2 (y )2 −xy + , 2 2 x y2 = The reader is invited to think about the case sin(2θ) = 0 geometrically. What happens to the axes in this case? 11.6 Hooked on Conics Again 831 2. From 16x2 + 24xy + 9...
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