AN80 Answers: Chapter 10 43. Hyperbola 45. Hyperbola 47. Parabola 49. Ellipse 51. Ellipse 53. 23.6° 55. Refer to equation (6): A A B C cos sin cos sin 2 2 θ θ θ θ ′ = + + B B C A C A B C D D E E D E F F cos sin 2 sin cos sin sin cos cos cos sin sin cos 2 2 2 2 θ θ θ θ θ θ θ θ θ θ θ θ ( ) ( )( ) ′ = − + − ′ = − + ′ = + ′ =− + ′ = 57. Use Problem 55 to find B AC 4 . 2′ − ′ ′ After much cancellation, B AC B AC 4 4 . 2 2 ′ − ′ ′ = − 59. The distance between P1 and P2 in the x y -plane ′ ′ equals x x y y . 2 1 2 2 1 2 ( ) ( ) ′ − ′ + ′ − ′ Assuming that x x y cos sin θ θ ′ = − and y x y sin cos , θ θ ′ = + then θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ′ − ′ = − − + = − − − − + − ′ − ′ = + − − = − + − − + − x x x y x y x x x x y y y y y y x y x y x x x x y y y y cos sin cos sin cos 2sin cos sin , and sin cos sin cos sin 2 sin cos cos . 2 1 2 2 2 1 1 2 2 2 1 2 2 1 2 1 2 2 1 2 2 1 2 2 2 1 1 2 2 2 1 2 2 1 2 1 2 2 1 2 Therefore, x x y y x x x x y y y y cos sin sin cos 2 1 2 2 1 2 2 2 1 2 2 2 1 2 2 2 1 2 2 2 1 2 θ θ θ θ ( ) ( ) ( ) ( ) ( ) ( ) − + − = − + − + − + − ′ ′ ′ ′ x x y y x x y y cos sin sin cos . 2 1 2 2 2 2 1 2 2 2 2 1 2 2 1 2 θ θ θ θ ( ) ( ) ( ) ( ) ( ) ( ) = − + + − + = − + − 63. A B C 39.4 , 54.7 , 85.9 ≈ ° ≈ ° ≈ ° 64. 38.5 65. r cos sin 1 2 θ θ = 66. i 29 cos291.8 sin291.8 ( ) ° + ° 67. x x x x 2 2 3 52 96 3 4 1 2 2 2 9 ( ) ( ) ( ) − + + + − 68. 5 7 69. 5{ } 70. 8.33 71. 8 72. y 5 = − 10.6 Assess Your Understanding (page 733) 3. conic; focus; directrix 4. parabola; ellipse; hyperbola 5. b 6. T 7. Parabola; directrix is perpendicular to the polar axis, 1 unit to the right of the pole. 9. Hyperbola; directrix is parallel to the polar axis, 4 3 units below the pole. 11. Ellipse; directrix is perpendicular to the polar axis, 3 2 units to the left of the pole. 13. Parabola; directrix is perpendicular to the polar axis, 1 unit to the right of the pole; vertex is at 1 2 , 0 . ( ) y x Directrix P 2 1, 3P 2 1, 1 2 , 0 Polar axis 2 2 15. Ellipse; directrix is parallel to the polar axis, 8 3 units above the pole; vertices are at 8 7 , 2 π ( ) and 8, 3 2 . π ( ) y x Polar axis Directrix 3P 2 8, (2, 0) (2, P) 8 7 P 2 , 2 5 17. Hyperbola; directrix is perpendicular to the polar axis, 3 2 units to the left of the pole; vertices are at 3, 0 ( ) − and 1, . π ( ) y x Polar axis Directrix (23, 0) P 2 3, 3P 2 3, (1, P) 5 5 37. 63 θ ≈ ° (see Problem 27) y x8 2′ = ′ Parabola Vertex at 0, 0 ( ) Focus at 2, 0 ( ) y 5 x x9 y9 5 (2, 0) 39. 34 θ ≈ ° (see Problem 29) x y 2 4 1 2 2 ( ) ′ − + ′ = Ellipse Center at 2, 0 ( ) Major axis is the x -axis. ′ Vertices at 4, 0 ( ) and 0, 0 ( ) y x 5 6 (2, 1) (4, 0) x9 y9 (2, 21) 41. cot 2 7 24 ; θ ( ) = sin 3 5 37 1 θ ( ) = ≈ ° − x y 1 6 1 6 2 ( ) ( ) ′ − = − ′ − Parabola Vertex at 1, 1 6 ( ) Focus at 1, 4 3 ( ) − y x 1, 2 4 3 1, 1 6 y9 x9 5 5 19. Ellipse; directrix is parallel to the polar axis, 8 units below the pole; vertices are at 8, 2 π ( ) and 8 3 , 3 2 . π ( ) y 8, P 2 (4, P) (4, 0) x 4 4 3P 2 , 8 3 Polar axis Directrix 21. Ellipse; directrix is parallel to the polar axis, 3 units below the pole; vertices are at 6, 2 π ( ) and 6 5 , 3 2 . π ( ) y x Polar axis Directrix (2, 0) 6, P 2 (2, P) 3P 2 , 6 5 7 5 23. Ellipse; directrix is perpendicular to the polar axis, 6 units to the left of the pole; vertices are at 6, 0 ( ) and 2, . π ( ) y x Polar axis Directrix (6, 0) 5 4 3P 2 3, P 2 3, (2, P)
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