1022 CHAPTER 10 Analytic Geometry Write an equation for each hyperbola. See Examples 5 and 6. 43. x-intercepts 1{3, 02; foci at 1-5, 02, 15, 02 44. y-intercepts 10, {122; foci at 10, -152, 10, 152 45. vertices at 10, 62, 10, -62; asymptotes y = { 1 2x 46. vertices at 1-10, 02, 110, 02; asymptotes y = {5x 47. vertices at 1-3, 02, 13, 02; passing through the point 1-6, -12 48. vertices at 10, 52, 10, -52; passing through the point 1-3, 102 49. foci at A0, 213B , A0, -213B ; asymptotes y = {5x 50. foci at A -325, 0B , A325, 0B ; asymptotes y = {2x 51. vertices at 14, 52, 14, 12; asymptotes y = {71x - 42 + 3 52. vertices at 15, -22, 11, -22; asymptotes y = { 3 21x - 32 - 2 53. center at 11, -2 ; focus at 1-2, -22; vertex at 1-1, -22 54. center at 19, -72; focus at 19, -172; vertex at 19, -13 55. e = 3; center at 10, 02; vertex at 10, 72 56. e = 5 3; center at 18, 72; focus at 13, 72 57. e = 25 9 ; foci at 19, -12, 1-11, -12 58. e = 5 4; vertices at 12, 102, 12, 22 Determine the two equations necessary to graph each hyperbola using a graphing calculator, and graph it in the viewing window indicated. See Example 1. 59. x2 4 - y2 16 = 1; 3-6.6, 6.64 by 3-8, 84 60. x2 25 - y2 49 = 1; 3-10, 104 by 3-10, 104 61. 4y2 - 36x2 = 144; 3-10, 104 by 3-15, 154 62. y2 - 9x2 = 9; 3-10, 104 by 3-10, 104 Solve each problem. 63. (Modeling) Atomic Structure In 1911, Ernest Rutherford discovered the basic structure of the atom by “shooting” positively charged alpha particles with a speed of 107 m per sec at a piece of gold foil 6 * 10-7 m thick. Only a small percentage of the alpha particles struck a gold nucleus head-on and were deflected directly back toward their source. The rest of the particles often followed a hyperbolic trajectory because they were repelled by positively charged gold nuclei. As a result of this famous experiment, Rutherford proposed that the atom was composed mostly of empty space with a small and dense nucleus. The figure on the next page shows an alpha particle A initially approaching a gold nucleus N and being deflected at an angle u = 90°. N is located at a focus of the hyperbola, and the trajectory of A passes through a vertex of the hyperbola. (Data from Semat, H., and J. Albright, Introduction to Atomic and Nuclear Physics, Fifth Edition, International Thomson Publishing.) Ernest Rutherford
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