728 CHAPTER 10 Analytic Geometry 58. Prove that, except for degenerate cases, the equation + + + + + = Ax Bxy Cy Dx Ey F 0 2 2 (a) Defines a parabola if − = B AC 4 0. 2 (b) Defines an ellipse (or a circle) if − < B AC 4 0. 2 (c) Defines a hyperbola if − > B AC 4 0. 2 59. Challenge Problem Use the rotation formulas (5) to show that distance is invariant under a rotation of axes. That is, show that the distance from ( ) = P x y , 1 1 1 to ( ) = P x y , 2 2 2 in the xy -plane equals the distance from P x y , 1 1 1 ( ) = ′ ′ to P x y , 2 2 2 ( ) = ′ ′ in the ′ ′ x y-plane. 60. Challenge Problem Show that the graph of the equation + = > x y a a , 0, 1 2 1 2 1 2 is part of the graph of a parabola. In Problems 55–58, apply the rotation formulas (5) to + + + + + = Ax Bxy Cy Dx Ey F 0 2 2 to obtain the equation ′′ + ′′′+ ′′ + ′′+ ′′+ ′= Ax Bxy Cy Dx Ey F 0 2 2 55. Express ′ ′ ′ ′ ′ A B C D E ,,, ,,and ′F in terms of A B C D E F , , , , , , and the angle θ of rotation. [ Hint : Refer to equation (6).] 56. Show that + = ′ + ′ A C A C, which proves that +A C is invariant ; that is, its value does not change under a rotation of axes. 57. Refer to Problem 56. Show that − B AC 4 2 is invariant. 61. Formulate a strategy for analyzing and graphing an equation of the form + + + + = Ax Cy Dx Ey F 0 2 2 62. Explain how your strategy presented in Problem 61 changes if the equation is of the form + + + + + = Ax Bxy Cy Dx Ey F 0 2 2 Explaining Concepts Retain Your Knowledge Problems 63–72 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. 63. Solve the triangle: = = a b 7, 9, and = c 11 64. Find the area of the triangle: = = = ° a b C 14, 11, 30 65. Transform the equation = xy 1 from rectangular coordinates to polar coordinates. 66. Write the complex number − i 2 5 in polar form. 67. Simplify ( ) ( ) ( ) ( ) ( ) − ⋅ + ⋅ − + ⋅ − ⋅ ⎡ ⎣ − ⎤⎦ x x x x x x 4 1 3 2 3 2 2 3 8 4 1 8 4 1 2 8 2 3 2 7 2 8 2 68. M varies directly as x and the square of d. If = M 90 when = x 14 and = d 3, find the proportionality constant. 69. Solve the equation ( ) + − = x x log log 4 1. 5 5 70. The graph of ( ) ( ) = + + − f x x x 25 1 2 8 2 has an absolute minimum when + − = x x 25 1 2 0. 2 What is the minimum value rounded to two decimal places? 71. If ( ) = − + g x x 7 2, find ( ) −g 3 . 1 72. Find the horizontal asymptote for the graph of ( ) = − + f x e4 5 x 1 ‘Are You Prepared?’ Answers 1. A B A B sin cos cos sin + 2. 2 sin cos θ θ 3. 1 cos 2 θ − 4. 1 cos 2 θ + 10.6 Polar Equations of Conics Now Work the ‘Are You Prepared?’ problems on page 733. • Polar Coordinates (Section 9.1, pp. 600–607) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Analyze and Graph Polar Equations of Conics (p. 728) 2 Convert the Polar Equation of a Conic to a Rectangular Equation (p. 733) 1 Analyze and Graph Polar Equations of Conics In Sections 10.2 through 10.4, we gave individual definitions for a parabola, ellipse, and hyperbola based on geometric properties and the distance formula. This section presents an alternative definition that simultaneously defines all these conics and is well suited to polar coordinate representation. (Refer to Section 9.1.)
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