SECTION 10.4 The Hyperbola 717 (e) Check the box “Show Equation of Hyperbola”. What is the equation of the hyperbola? • x y 9 7 1 2 2 − = • x y 7 9 1 2 2 − = • y x 9 7 1 2 2 − = • y x 7 9 1 2 2 − = (f) Hit the reset button ( ) on the interactive figure (upperright corner). Move the point F1 to 0, 5 ( ) − and move the point F2 to 0, 5 . ( ) Set the value of a to 3. Check the box “Show Hyperbola”. What is the value of c, the distance from the center to either focus? (g) Click “Show Equation of Hyperbola”. What is the equation of the hyperbola? • x y 16 9 1 2 2 − = • x y 9 16 1 2 2 − = • y x 16 9 1 2 2 − = • y x 9 16 1 2 2 − = (h) If the graph of a hyperbola opens left and right, then the -term must be first. • x2 • y2 13. 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). Check the boxes “Show Hyperbola” and “Show Equation of Hyperbola”. (a) Set the value of a to 2. Move F1 to 2, 2 ( ) and F2 to 2, 4. ( ) − Look at the equation of the hyperbola and note the center. The transverse axis is the line through the vertices, foci, and center. What is the equation of the transverse axis? (b) Move F1 to 4, 4. ( ) − − Look at the equation of the hyperbola and note the center. What is the value of c, the distance from the center to either focus? (c) Leave the value of a at 2. Move F1 to 6, 1 ( ) − − and F2 to 2, 1. ( ) − Look at the equation of the hyperbola and note the center. If the transverse axis is parallel to the x-axis, then the equation of the hyperbola is of the form • x h a y k b 1 2 2 2 2 ( ) ( ) − − − = • y k a x h b 1 2 2 2 2 ( ) ( ) − − − = (d) If the transverse axis is parallel to the y-axis, then the equation of the hyperbola is of the form • x h a y k b 1 2 2 2 2 ( ) ( ) − − − = • y k a x h b 1 2 2 2 2 ( ) ( ) − − − = 14. Multiple Choice In a hyperbola, if a 3 = and c 5, = then b = . (a) 1 (b) 2 (c) 4 (d) 8 15. For the hyperbola x y 4 9 1, 2 2 − = the value of a is , the value of b is , and the transverse axis is the -axis. 16. For the hyperbola y x 16 81 1, 2 2 − = the asymptotes are and . Skill Building In Problems 17–20, the graph of a hyperbola is given. Match each graph to its equation. (A) x y 4 1 2 2 − = (B) x y 4 1 2 2 − = (C) y x 4 1 2 2 − = (D) y x 4 1 2 2 − = 17. x y 23 3 3 23 18. x y 24 4 4 24 19. x y 24 4 4 24 20. x y 23 3 3 23 In Problems 21–30, find an equation for the hyperbola described. Graph the equation. 21. Center at 0, 0 ; ( ) focus at 3, 0 ; ( ) vertex at 1, 0 ( ) 22. Center at 0, 0 ; ( ) focus at 0, 5 ; ( ) vertex at 0, 3 ( ) 23. Center at 0, 0 ; ( ) focus at 0, 6 ; ( ) − vertex at 0, 4 ( ) 24. Center at 0, 0 ; ( ) focus at 3, 0 ; ( ) − vertex at 2, 0 ( ) 25. Foci at 5, 0 ( ) − and 5,0 ; ( ) vertex at 3, 0 ( ) 26. Focus at 0, 6 ; ( ) vertices at 0, 2 ( ) − and 0, 2 ( ) 27. Vertices at 0, 6 ( ) − and 0,6 ; ( ) asymptote the line y x2 = 28. Vertices at 4, 0 ( ) − and 4,0 ; ( ) asymptote the line y x2 = 29. Foci at 4, 0 ( ) − and 4,0 ; ( ) asymptote the line y x = − 30. Foci at 0, 2 ( ) − and 0,2 ; ( ) asymptote the line y x = − In Problems 31–38, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation. 31. x y 25 9 1 2 2 − = 32. y x 16 4 1 2 2 − = 33. x y 4 16 2 2 − = 34. y x 4 16 2 2 − = 35. y x9 9 2 2 − = 36. x y 4 2 2 − = 37. y x 25 2 2 − = 38. x y 2 4 2 2 − =
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