720 CHAPTER 10 Analytic Geometry Retain Your Knowledge Problems 91–99 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. 91. For f x x 1 2 sin 3 5, π ( ) ( ) = − + + find the amplitude, period, phase shift, and vertical shift. 92. Solve the triangle described: a b 7, 10, = = and C 100 = ° 93. Find the rectangular coordinates of the point with the polar coordinates 12, 3 . π ( ) − 94. Transform the polar equation r 6 sinθ = to an equation in rectangular coordinates. Then identify and graph the equation. 95. What is the inverse function for f x e3 4? x 1 ( ) = + − 96. Find the area of the region enclosed by the graphs of = − = + y x y x 9 and 3. 2 97. Solve x x x x 2 3 5 2 1. 2 2 ( ) ( ) + + = + + 98. Find the midpoint of the line segment connecting the points 3, 8 and 2, 5 . ( ) ( ) − − 99. Evaluate ( ) ( ) − x cos sin 4 . 1 ‘Are You Prepared?’ Answers 1. 5 2 2. 25 4 3. 0, 3, 0,3 ( ) ( ) − 4. True 5. right; 5; down; 4 6. Vertical: x x 2, 2; = − = horizontal: y 1 = 10.5 Rotation of Axes; General Form of a Conic Now Work the ‘Are You Prepared?’ problems on page 726. • Sum Formulas for Sine and Cosine (Section 7.5, pp. 511 and 514) • Half-angle Formulas for Sine and Cosine (Section 7.6, p. 528) • Double-angle Formulas for Sine and Cosine (Section 7.6, p. 525) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Identify a Conic (p. 720) 2 Use a Rotation of Axes to Transform Equations (p. 721) 3 Analyze an Equation Using a Rotation of Axes (p. 724) 4 Identify Conics without Rotating the Axes (p. 726) In this section, we show that the graph of a general second-degree polynomial equation containing two variables x and y —that is, an equation of the form + + + + + = Ax Bxy Cy Dx Ey F 0 2 2 (1) where A B , , and C are not all 0—is a conic.We are not concerned with the degenerate cases of equation (1), such as + = x y 0, 2 2 whose graph is a single point ( ) 0, 0 ; or + + = x y3 3 0, 2 2 whose graph contains no points; or − = x y4 0, 2 2 whose graph is two lines, − = x y2 0 and + = x y2 0. 1 Identify a Conic We begin with the case where = B 0. In this case, the term containing xy is not present, so equation (1) has the form + + + + = Ax Cy Dx Ey F 0 2 2 (2) where either ≠ A 0 or ≠ C 0. We have already discussed how to identify the graph of an equation of this form. We complete the squares of the quadratic expressions in x or y, or both. Then the conic can be identified by comparing it to one of the forms studied in Sections 10.2 through 10.4. But the conic can be identified directly from its equation without completing the squares.
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