726 CHAPTER 10 Analytic Geometry ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 10.5 Assess Your Understanding 1. The sum formula for the sine function is A B sin ( ) + = . (p. 514) 2. The Double-angle Formula for the sine function is θ ( ) = sin 2 . (p. 525) 3. If θ is acute, the Half-angle Formula for the sine function is sin 2 θ = . (p. 528) 4. If θ is acute, the Half-angle Formula for the cosine function is cos 2 θ = . (p. 528) 4 Identify Conics without Rotating the Axes Suppose that we are required only to identify (rather than analyze) the graph of an equation of the form + + + + + = ≠ Ax Bxy Cy Dx Ey F B 0 0 2 2 (8) Applying the rotation formulas (5) to this equation gives an equation of the form ′′ + ′′′+ ′′ + ′′+ ′′+ ′= Ax Bxy Cy Dx Ey F 0 2 2 (9) where ′ ′ ′ ′ ′ A B C D E ,,, ,,and ′ F can be expressed in terms of A B C D E F , , , , , and the angle θ of rotation (see Problem 55). It can be shown that the value of − B AC 4 2 in equation (8) and the value of ′ − ′ ′ B AC 4 2 in equation (9) are equal no matter what angle θ of rotation is chosen (see Problem 57). In particular, if the angle θ of rotation satisfies equation (7), then ′ = B 0 in equation (9), and − = − ′ ′ B AC AC 4 4 . 2 Since equation (9) then has the form of equation (2), ′′ + ′′ + ′′+ ′′+ ′= Ax Cy Dx Ey F 0 2 2 we can identify its graph without completing the squares, as we did in the beginning of this section. In fact, now we can identify the conic described by any equation of the form of equation (8) without rotating the axes. THEOREM Identifying Conics without Rotating the Axes Except for degenerate cases, the equation + + + + + = Ax Bxy Cy Dx Ey F 0 2 2 (10) • Defines a parabola if − = B AC 4 0. 2 • Defines an ellipse (or a circle) if − < B AC 4 0. 2 • Defines a hyperbola if − > B AC 4 0. 2 You are asked to prove this theorem in Problem 58. Because of the above theorem, equation (10) is called the general equation of a conic . Identifying a Conic without Rotating the Axes Identify the graph of the equation − + − − − = x xy y x y 8 12 17 4 5 2 5 15 0. 2 2 Solution EXAMPLE 5 Here = = − A B 8, 12, and = C 17, so − = − B AC 4 400. 2 Since − < B AC 4 0, 2 the equation defines an ellipse. Now Work PROBLEM 43 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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