Section 10.3 AN75 85. 0,2, 0, 2, 36,0; ( ) ( ) ( ) − − symmetric with respect to the x-axis. 86. 5{ } 87. sin 5 89 89 ; cos 8 89 89 ; csc 89 5 ; sec 89 8 ; θ θ θ θ = = − = = − cot 8 5 θ = − 88. 2 10 3 − 89. 11 13 6 90. x y 12 7 6 2 2 ( ) ( ) + + − = 91. ln2 4 92. cos17° 93. 1, 2, 3, 6 { } − 10.3 Assess Your Understanding (page 701) 7. ellipse 8. b 9. (a) 2, 3 ( ) (b) 10 (c) 5 (d) 4 (e) 3 (f) a c b 2 2 2 − = (g) 5; 3; 4 (h) x y 16 25 1; 2 2 + = The denominator of the x -term 2 is b2 and the denominator of the y -term 2 is a .2 10. (a) 5; 3; parallel to x-axis (b) 5; 4; parallel to y-axis (c) x h a y k b 1 2 2 2 2 ( ) ( ) − + − = 11. 0, 5; 0,5 ( ) ( ) − 12. 5; 3; x 13. 2, 3; 6, 3 ( ) ( ) − − − 14. a 15. C 17. B 19. Vertices: 5, 0 , 0, 5 ( ) ( ) − Foci: 21, 0 , 21, 0 ( ) ( ) − y 5 x 5 (0, 2) (5, 0) (25, 0) (0, 22) (2!21, 0) (!21, 0) 21. Vertices: 0, 5 , 0, 5 ( ) ( ) − Foci: 0, 4 , 0, 4 ( ) ( ) − y 5 x 5 (0, 4) (0, 5) (3, 0) (0, 24) (23, 0) (0, 25) 23. x y 4 16 1 2 2 + = Vertices: 0, 4 , 0, 4 ( ) ( ) − Foci: 0, 2 3 , 0,2 3 ( ) ( ) − y 5 x 5 (0, 24) (22, 0) (0, 4) (2, 0) (0, 2 3) ! (0, 22 3) ! 25. x y 8 2 1 2 2 + = Vertices: 2 2,0, 2 2,0 ( ) ( ) − Foci: 6, 0 , 6, 0 ( ) ( ) − y 5 x 5 (0, ) !2 (0, 2 ) !2 (2 , 0) !2 ( , 0) !6 (2 , 0) !6 (22 , 0) !2 47. Vertex: 3, 1 ; ( ) − focus: 3, 5 4 ; ( ) − directrix: y 3 4 = − y 1 x 3 V 5 (3, 21) 5 4 3, 2 F 5 3 4 D: y 5 2 49. Vertex: 2, 3 ; ( ) − focus: 4, 3; ( ) − directrix: x 0 = y 2 x 8 V 5 (2, 23) F 5 (4, 23) (4, 27) (4, 1) D: x 5 0 51. Vertex: 0, 2 ; ( ) focus: 1, 2 ; ( ) − directrix: x 1 = y 8 x 2 (21, 0) D: x 5 1 F 5 (21, 2) V 5 (0, 2) (21, 4) 53. Vertex: 4, 2; ( ) − − focus: 4, 1; ( ) − − directrix: y 3 = − y 5 x 1 (26, 21) (22, 21) V 5 (24, 22) F 5 (24, 21) D: y 5 23 81. Cy Dx C D Cy Dx y D C x 0, 0, 0 2 2 2 + = ≠ ≠ = − = − This is the equation of a parabola with vertex at 0, 0 ( ) and axis of symmetry the x-axis. The focus is D C4 , 0 ; ( ) − the directrix is the line x D C4 . = The parabola opens to the right if D C 0 − > and to the left if D C 0. − < 83. Cy Dx Ey F C Cy Ey Dx F y E C y D C x F C y E C D C x F C E C y E C D C x E CF C 0, 0 2 4 2 4 4 2 2 2 2 2 2 2 2 2 ( ) ( ) + + + = ≠ + = − − + = − − + = − − + + = − + − (a) If D 0, ≠ then the equation may be written as y E C D C x E CF CD 2 4 4 . 2 2 ( ) ( ) + = − − − This is the equation of a parabola with vertex at E CF CD E C 4 4 , 2 2 ( ) − − and axis of symmetry parallel to the x-axis. (b)–(d) If D 0, = the graph of the equation contains no points if E CF 4 0, 2 − < is a single horizontal line if E CF 4 0, 2 − = and is two horizontal lines if E CF 4 0. 2 − > 55. Vertex: 1, 1; ( ) − − focus: 3 4 , 1; ( ) − − directrix: x 5 4 = − y 2.5 x 2.5 5 4 D: x 5 2 V 5 (21, 1) F 5 3 4 , 2 21 3 , 4 1 2 2 2 3 , 4 3 2 2 2 59. y x 1 2 ( ) − = 61. y x 1 2 2 ( ) ( ) − = − − 63. x y 4 1 2 ( ) = − 65. y x 1 2 2 2 ( ) = + 67. 20 ft 69. 24.31 ft, 18.75 ft, 7.64 ft 71. 1.5625 ft from the base of the dish, along the axis of symmetry 73. 1 in. from the vertex, along the axis of symmetry 75. 0.78125 ft 77. 4.17 in. from the base, along the axis of symmetry 79. (a) y x 2 315 630 2 = − + (b) 567 ft: 119.7 ft; 478 ft: 267.3 ft; 308 ft: 479.4 ft (c) No 57. Vertex: 2, 8 ; ( ) − focus: 2, 31 4 ; ( ) − directrix: y 33 4 = − y 1 x 8 V 5 (2, 28) 31 4 2, 2 F 5 33 4 D: y 5 2
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