716 CHAPTER 10 Analytic Geometry Check: The difference between the distance from 2640, 12122 ( ) to the person at the point B 2640, 0 ( ) = − and the distance from 2640, 12122 ( ) to the person at the point A 2640, 0 ( ) = should be 1100. Using the distance formula, the difference of the distances is 2640 2640 12,122 0 2640 2640 12,122 0 1100 2 2 2 2 ( ) [ ] ( ) ( ) ( ) − − + − − − + − = as required. ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 10.4 Assess Your Understanding 1. The distance d from P 3, 4 1 ( ) = − to P 2, 1 2 ( ) = − is d = . (p. 14) 2. To complete the square of x x5 , 2 + add . (p. A29) 3. Find the intercepts of the equation y x 9 4 . 2 2 = + (pp. 20–21) 4. True or False The equation y x 9 2 2 = + is symmetric with respect to the x -axis, the y -axis, and the origin. (pp. 21–23) 5. To graph y x 5 4, 3 ( ) = − − shift the graph of y x3 = to the left right ( ) unit(s) and up down ( ) unit(s). (pp. 112–120) 6. Find the vertical asymptotes, if any, and the horizontal or oblique asymptote, if any, of y x x 9 4 . 2 2 = − − (pp. 238–241) Concepts and Vocabulary 7. A(n) is the collection of points in a plane, the difference of whose distances from two fixed points is a constant. 8. For a hyperbola, the foci lie on a line called the . Answer Problems 9–11 using the figure below. x y V2 V1 (h, k) Transverse axis F2 F1 9. Multiple Choice The equation of the hyperbola is of the form (a) x h a y k b 1 2 2 2 2 ( ) ( ) − − − = (b) y k a x h b 1 2 2 2 2 ( ) ( ) − − − = (c) x h a y k b 1 2 2 2 2 ( ) ( ) − + − = (d) x h b y k a 1 2 2 2 2 ( ) ( ) − + − = 10. If the center of the hyperbola is 2, 1 ( ) and a 3, = then the coordinates of the vertices are and . 11. If the center of the hyperbola is 2, 1 ( ) and c 5, = then the coordinates of the foci are and . 12. Interactive Figure Exercise Exploring the Graph of a Hyperbola Open the “Hyperbola” interactive figure, which is available in the Video & Resource Library of MyLab Math (under Sullivan Interactive Figures). (a) Set the slider to a 2. = Grab the point F,1 which represents a focus of a hyperbola, and move the point F1 to 2, 1 . ( ) − Grab the point F ,2 which represents the other focus of a hyperbola, and move the point F2 to 4, 3. ( ) − What is the center of the hyperbola? Write your answer as an ordered pair. Confirm the center of the hyperbola is at the midpoint of the two foci. (b) Click “Show Trace”. Grab Point A and move it around the Cartesian Plane. Complete a trace of the entire hyperbola (be sure your hyperbola has two branches). What is the absolute value of the difference of the distances from the foci to any point on the hyperbola? (c) Hit the reset button ( ) on the interactive figure (upper-right corner). Move the point F1 to 4, 0 ( ) − and move the point F2 to 4, 0 . ( ) Click “Show Hyperbola”. Use the a slider to change the value of a from 2 to 3. What happens to the vertices as a changes? What does the value of a represent? Notice that the absolute value of the difference of the distances from the foci to any point on the hyperbola is a2 . Click “Show Trace” to verify this. (d) What is the value of c, the distance from the center to either focus? 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure Now Work PROBLEM 77 (continued)
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