Unformatted text preview: tion 7.5, so we proceed here
under the assumption that B = 0. We rotate the xy axes counterclockwise through an angle
θ which satisﬁes cot(2θ) = A−C to produce an equation with no x y term in accordance with
B
Theorem 11.10: A (x )2 + C (y )2 + Dx + Ey + F = 0. In this form, we can invoke Exercise 10
in Section 7.5 once more using the product A C . Our goal is to ﬁnd the product A C in terms of
the coeﬃcients A, B and C in the original equation. To that end, we make the usual substitutions
x = x cos(θ) − y sin(θ) y = x sin(θ) + y cos(θ) into Ax2 + Bxy + Cy 2 + Dx + Ey + F = 0. We
leave it to the reader to show that, after gathering like terms, the coeﬃcient A on (x )2 and the
coeﬃcient C on (y )2 are
A = A cos2 (θ) + B cos(θ) sin(θ) + C sin2 (θ) C = A sin2 (θ) − B cos(θ) sin(θ) + C cos2 (θ) In order to make use of the condition cot(2θ) = A−C , we rewrite our formulas for A and C using
B
the power reduction formulas. After some regrouping, we get
2A = [(A + C...
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 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

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