Section 10.5 AN79 77. The fireworks display is 50,138 ft north of the person at point A. 79. The tower is 592.4 ft tall. 81. (a) y x = ± (b) x y x 100 100 1, 0 2 2 − = ≥ 83. y x 81 88 1 2 2 − = 85. If the eccentricity is close to 1, the “opening” of the hyperbola is very small. As e increases, the opening gets bigger. 87. x y 4 1; 2 2 − = asymptotes y x 1 2 = ± y x 4 1; 2 2 − = asymptotes y x 1 2 = ± y 2.5 x 1.5 (2, 0) (0, 1) (0, 21) (22, 0) x2 4 2 y2 5 1 x2 4 5 1 y2 2 1 2 y 5 2 x 1 2 y 5 x 89. Ax Cy F Ax Cy F 0 2 2 2 2 + + = + = − If A and C are of opposite sign and F 0, ≠ this equation may be written as x F A y F C 1, 2 2 ( ) ( ) − + − = where F A − and F C − are opposite in sign. This is the equation of a hyperbola with center at 0, 0 . ( ) The transverse axis is the x-axis if F A 0; − > the transverse axis is the y-axis if F A 0. − < 91. Amplitude 1 2 ; Period 2 3 ; Phaseshift 3 π π = = = − ; Vertical shift 5 = 10.5 Assess Your Understanding (page 726) 5. A C B cot 2θ ( ) = − 6. d 7. B AC 4 0 2 − < 8. c 9. T 10. F 11. Parabola 13. Ellipse 15. Hyperbola 17. Hyperbola 19. Circle 21. x x y y x y 2 2 , 2 2 ( ) ( ) = ′ − ′ = ′ + ′ 23. x x y y x y 2 2 , 2 2 ( ) ( ) = ′ − ′ = ′ + ′ 25. x x y y x y 1 2 3 , 1 2 3 ( ) ( ) = ′ − ′ = ′ + ′ 27. x x y y x y 5 5 2 , 5 5 2 ( ) ( ) = ′ − ′ = ′ + ′ 29. x x y y x y 13 13 3 2 , 13 13 2 3 ( ) ( ) = ′ − ′ = ′ + ′ 3P 4 U 5 5P 4 U 5 3P 2 U 5 7P 4 U 5 P 2 U 5 P 4 U 5 y x U 5 P U 5 0 1 2 3 4 5 6 94. x y 3 9; 2 2 ( ) + − = circle, radius 3, center at 0, 3 ( ) in rectangular coordinates 92. c A B 13.16, 31.6 , 48.4 ≈ ≈ ° = ° 93. 6, 63 ( ) − 95. f x x ln 4 3 1 1 ( ) ( ) = − + − 96. 9 4 9 2 2.57 squnits π − ≈ 97. 4 { } − 98. 1 2 , 3 2 ( ) − 99. x 16 4 2 − 31. 45 θ = ° (see Problem 21) x y 3 1 2 2 ′ − ′ = Hyperbola Center at origin Transverse axis is the x -axis. ′ Vertices at 1, 0 ( ) ± y x (21, 0) (1, 0) 2.5 x9 y9 2.5 33. 45 θ = ° (see Problem 23) x y 4 1 2 2 ′ + ′ = Ellipse Center at 0, 0 ( ) Major axis is the y -axis. ′ Vertices at 0, 2 ( ) ± y x 2.5 2.5 (0, 22) (0, 2) x9 y9 35. 60 θ = ° (see Problem 25) x y 4 1 2 2 ′ + ′ = Ellipse Center at 0, 0 ( ) Major axis is the x -axis. ′ Vertices at 2, 0 ( ) ± y x x9 2.5 (2, 0) (22, 0) 2.5 y9 71. Vertex: 0, 3 ; ( ) focus: 0, 7 ; ( ) directrix: y 1 = − y 8 x 10 V 5 (0, 3) F 5 (0, 7) D: y 5 21 73. x y 5 9 25 1 2 2 ( ) − + = Center: 5, 0 ; ( ) vertices: 5, 5 , 5, 5 ; ( ) ( ) − foci: 5, 4 , 5, 4 ( ) ( ) − y 5 x 9 (5, 0) (2, 0) (8, 0) V2 5 (5, 25) F2 5 (5, 24) V1 5 (5, 5) F1 5 (5, 4) 75. x y 3 8 5 2 ( ) ( ) − = + Vertex: 3, 5; ( ) − focus: 3, 3; ( ) − directrix: y 7 = − y 2 x 10 D: y 5 27 V 5 (3, 25) F 5 (3, 23)
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