752 CHAPTER 10 Analytic Geometry The Chapter Test Prep Videos include step-by-step solutions to all chapter test exercises. These videos are available in MyLab™ Math. In Problems 1–3, identify each equation. If it is a parabola, give its vertex, focus, and directrix; if an ellipse, give its center, vertices, and foci; if a hyperbola, give its center, vertices, foci, and asymptotes. 1. x y 1 4 9 1 2 2 ( ) + − = 2. y x 8 1 4 2 ( ) = − − 3. x y x y 2 3 4 6 13 2 2 + + − = In Problems 4–6, find an equation of the conic described; graph the equation. 4. Parabola: focus 1, 4.5 , ( ) − vertex 1, 3 ( ) − 5. Ellipse: center 0, 0 , ( ) vertex 0, 4, ( ) − focus 0, 3 ( ) 6. Hyperbola: center 2, 2 , ( ) vertex 2, 4 , ( ) contains the point 2 10, 5 ( ) + In Problems 7–9, identify each conic without completing the square or rotating axes. 7. x xy y x 2 5 3 3 7 0 2 2 + + + − = 8. x xy y y 3 2 3 1 0 2 2 − + + + = 9. x xy y x y 6 9 2 3 2 0 2 2 − + + − − = 10. Given the equation x xy y 41 24 34 25 0, 2 2 − + − = rotate the axes so that there is no xy-term.Analyze and graph the new equation. 11. Identify the conic represented by the polar equation r 3 1 2cos . θ = − Find the rectangular equation. 12. Graph the plane curve whose parametric equations are given, and show its orientation. Find the rectangular equation for the plane curve. x t t y t t t 3 2 1 0 9 ( ) ( ) = − = − ≤ ≤ 13. A parabolic reflector (paraboloid of revolution) is used by TV crews at football games to pick up the referee’s announcements, quarterback signals, and so on. A microphone is placed at the focus of the parabola. If a certain reflector is 4 feet wide and 1.5 feet deep, where should the microphone be placed? Chapter Test 1. For f x x x 3 5 2, 2 ( ) = − + − find f x h f x h h 0 ( ) ( ) + − ≠ 2. In the complex number system, solve the equation x x x x 9 33 71 57 10 0 4 3 2 + − − − = 3. For what numbers x is x x 6 ? 2 − ≥ 4. (a) Find the domain and range of y 3 2. x = + (b) Find the inverse of y 3 2 x = + and state its domain and range. 5. f x x log 2 4 ( ) ( ) = − (a) Solve f x 2. ( ) = (b) Solve f x 2. ( ) ≤ Cumulative Review 6. Find an equation for each of the following graphs. (a) Line: x y 2 1 –2 (b) Circle: x y 2 2 –1 4 (c) Ellipse: x y 2 3 –3 –2 (d) Parabola: x y 2 1 –1 (e) Hyperbola: x y 2 2 –2 –2 (3, 2) (f) Exponential: (1, 4) x y (0, 1) (21, ) 1 – 4
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