702 CHAPTER 10 Analytic Geometry Skill Building In Problems 15–18, the graph of an ellipse is given. Match each graph to its equation. (A) x y 4 1 2 2 + = (B) x y 4 1 2 2 + = (C) x y 16 4 1 2 2 + = (D) x y 4 16 1 2 2 + = 15. x y 2 22 4 24 16. x y 4 24 2 22 17. x y 3 23 23 3 18. x y 3 23 23 3 In Problems 19–28, analyze each equation. That is, find the center, vertices, and foci of each ellipse and graph it. 19. x y 25 4 1 2 2 + = 20. x y 9 4 1 2 2 + = 21. x y 9 25 1 2 2 + = 22. x y 16 1 2 2 + = 23. x y 4 16 2 2 + = 24. x y9 18 2 2 + = 25. y x 4 8 2 2 + = 26. y x 4 9 36 2 2 + = 27. x y 16 2 2 + = 28. x y 4 2 2 + = In Problems 29–40, find an equation for each ellipse. Graph the equation. 29. Center at 0, 0 ; ( ) focus at 3, 0 ; ( ) vertex at 5, 0 ( ) 30. Center at 0, 0 ; ( ) focus at 1, 0 ; ( ) − vertex at 3, 0 ( ) 31. Center at 0, 0 ; ( ) focus at 0, 4 ; ( ) − vertex at 0, 5 ( ) 32. Center at 0, 0 ; ( ) focus at 0, 1 ; ( ) vertex at 0, 2 ( ) − 33. Foci at 2, 0 ; ( ) ± length of the major axis is 6 34. Foci at 0, 2 ; ( ) ± length of the major axis is 8 35. Focus at 4, 0 ; ( ) − vertices at 5, 0 ( ) ± 36. Focus at 0, 4 ; ( ) − vertices at 0, 8 ( ) ± 37. Foci at 0, 3 ; ( ) ± x-intercepts are 2± 38. Vertices at 4, 0 ; ( ) ± y-intercepts are 1± 39. Center at 0, 0 ; ( ) vertex at ( ) = b 0, 4 ; 1 40. Vertices at c 5, 0 ; 2 ( ) ± = In Problems 45–56, analyze each equation; that is, find the center, foci, and vertices of each ellipse. Graph each equation. 45. x y 3 4 1 9 1 2 2 ( ) ( ) − + + = 46. x y 4 9 2 4 1 2 2 ( ) ( ) + + + = 47. x y 5 4 4 16 2 2 ( ) ( ) + + − = 48. x y 9 3 2 18 2 2 ( ) ( ) − + + = 49. x x y y 4 4 8 4 0 2 2 + + − + = 50. x y y 3 12 9 0 2 2 + − + = 51. x y x y 2 3 8 6 5 0 2 2 + − + + = 52. x y x y 4 3 8 6 5 2 2 + + − = 53. x y x y 9 4 18 16 11 0 2 2 + − + − = 54. x y x y 9 6 18 9 0 2 2 + + − + = 55. x y y 4 4 0 2 2 + + = 56. x y x 9 18 0 2 2 + − = In Problems 41–44, find an equation for each ellipse. 41. x y 3 23 23 3 (21, 1) 42. (21, 21) x y 3 23 23 3 43. x y 3 23 23 3 (1, 0) 44. (0, 1) x y 3 23 23 3 In Problems 57–66, find an equation for each ellipse. Graph the equation. 57. Center at 2, 2 ; ( ) − vertex at 7, 2 ; ( ) − focus at 4, 2 ( ) − 58. Center at 3, 1 ; ( ) − vertex at 3, 3 ; ( ) − focus at 3, 0 ( ) − 59. Vertices at 4, 3 ( ) and 4,9 ; ( ) focus at 4, 8 ( ) 60. Foci at 1, 2 ( ) and 3, 2 ; ( ) − vertex at 4, 2 ( ) − 61. Foci at 5, 1 ( ) and 1, 1 ; ( ) − length of the major axis is 8 62. Vertices at 2, 5 ( ) and c 2, 1; 2 ( ) − = 63. Center at 1, 2 ; ( ) focus at 4, 2 ; ( ) contains the point 1, 3 ( ) 64. Center at 1, 2 ; ( ) focus at 1, 4 ; ( ) contains the point 2, 2 ( ) 65. Center at 1, 2 ; ( ) vertex at 4, 2 ; ( ) contains the point 1, 5 ( ) 66. Center at 1, 2 ; ( ) vertex at 1, 4 ; ( ) contains the point 1 3, 3 ( ) + In Problems 67–70, graph each function. Be sure to label all the intercepts. [Hint: Notice that each function is half an ellipse.] 67. f x x 16 4 2 ( ) = − 68. f x x 9 9 2 ( ) = − 69. f x x 64 16 2 ( ) = − − 70. f x x 4 4 2 ( ) = − −
RkJQdWJsaXNoZXIy NjM5ODQ=