718 CHAPTER 10 Analytic Geometry In Problems 39–42, find an equation for each hyperbola. 39. x y 23 3 3 23 y 5 2x y 5 x 40. x y 23 3 3 23 y 5 2x y 5 x 41. x y 25 5 10 210 y 5 2x y 5 22x 42. x y 25 5 5 25 y 5 2x y 5 22x In Problems 43–50, find an equation for the hyperbola described. Graph the equation. 43. Center at 4, 1 ; ( ) − focus at 7, 1 ; ( ) − vertex at 6, 1 ( ) − 44. Center at 3, 1 ; ( ) − focus at 3, 6 ; ( ) − vertex at 3, 4 ( ) − 45. Center at 3, 4; ( ) − − focus at 3, 8; ( ) − − vertex at 3, 2 ( ) − − 46. Center at 1, 4 ; ( ) focus at 2, 4 ; ( ) − vertex at 0, 4 ( ) 47. Foci at 3, 7 ( ) and 7,7 ; ( ) vertex at 6, 7 ( ) 48. Focus at 4, 0 ; ( ) − vertices at 4, 4 ( ) − and 4, 2 ( ) − 49. Vertices at 1, 1 ( ) − − and 3, 1; ( ) − asymptote the line ( ) + = − y x 1 3 2 1 50. Vertices at 1, 3 ( ) − and 1,1 ; ( ) asymptote the line ( ) + = − y x 1 3 2 1 In Problems 51–64, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation. 51. x y 2 4 3 9 1 2 2 ( ) ( ) − − + = 52. y x 3 4 2 9 1 2 2 ( ) ( ) + − − = 53. y x 2 4 2 4 2 2 ( ) ( ) − − + = 54. x y 4 9 3 9 2 2 ( ) ( ) + − − = 55. x y 1 2 4 2 2 ( ) ( ) + − + = 56. y x 3 2 4 2 2 ( ) ( ) − − + = 57. x y x y 2 2 1 0 2 2 − − − − = 58. y x y x 4 4 1 0 2 2 − − + − = 59. y x y x 4 4 8 4 0 2 2 − − − − = 60. x y x y 2 4 4 4 0 2 2 − + + − = 61. x y x y 4 24 4 16 0 2 2 − − − + = 62. y x x y 2 2 8 3 0 2 2 − + + + = 63. y x x y 4 16 2 19 0 2 2 − − − − = 64. x y x y 3 8 6 4 0 2 2 − + − + = In Problems 65–68, graph each function. Be sure to label any intercepts. [Hint: Notice that each function is half a hyperbola.] 65. f x x 16 4 2 ( ) = + 66. f x x 9 9 2 ( ) = − + 67. f x x 25 2 ( ) = − − + 68. f x x 1 2 ( ) = − + Mixed Practice In Problems 69–76, analyze each equation. 69. x y 3 4 25 1 2 2 ( ) − − = 70. y x 2 16 2 4 1 2 2 ( ) ( ) + − − = 71. x y 16 3 2 ( ) = − 72. y x 12 1 2 ( ) = − + 73. x y x 25 9 250 400 0 2 2 + − + = 74. x y x y 36 2 288 541 0 2 2 + − + + = 75. x x y 6 8 31 0 2 − − − = 76. x y x y 9 18 8 88 0 2 2 − − − − = Applications and Extensions 77. Fireworks Display Suppose that two people standing 2 miles apart both see the burst from a fireworks display. After a period of time the first person, standing at point A, hears the burst. One second later the second person, standing at point B, hears the burst. If the person at point B is due west of the person at point A, and if the display is known to occur due north of the person at point A, where did the fireworks display occur? 78. Lightning Strikes Suppose that two people standing 1 mile apart both see a flash of lightning. After a period of time the first person, standing at point A, hears the thunder. Two seconds later the second person, standing at point B, hears the thunder. If the person at point B is due west of the person at point A, and if the lightning strike is known to occur due north of the person standing at point A, where did the lightning strike occur? 79. Nuclear Power Plant Some nuclear power plants utilize “natural draft” cooling towers in the shape of a hyperboloid, a solid obtained by rotating a hyperbola about its conjugate axis. Suppose that such a cooling tower has a base diameter of 400 feet, and the diameter at its narrowest point, 360 feet above the ground, is 200 feet. If the diameter at the top of the tower is 300 feet, how tall is the tower? Source: Bay Area Air Quality Management District 80. An Explosion Two recording devices are set 2400 feet apart, with the device at point A to the west of the device at point B. At a point between the devices 300 feet from point B, a small amount of explosive is detonated. The recording devices record the time until the sound reaches each. How far directly north of point B should a second explosion be done so that the measured time difference recorded by the devices is the same as that for the first detonation?
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