ISM_T11_C10_I - 680 Chapter 10 Conic Sections and Polar...

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680 Chapter 10 Conic Sections and Polar Coordinates 50. r 1 sin and r 1 sin 1 sin 1 sin œ± œ² Ê ± )) ) ) 2 sin 0 sin 0 0, ; 0 or Êœ Ê œ Ê œ œ ) 1 ) 1 r 1. The points of intersection are (1 0) and (1 ). ß ß 1 The point of intersection ( 0) is found by graphing. 51. r 1 sin and r 1 sin intersect at all points of œ²± r 1 sin because the graphs coincide. This can be ) seen by graphing them. 52. r 1 cos and r 1 cos intersect at all points of r 1 cos because the graphs coincide. This can be ) seen by graphing them. 53. r sec and r 2 sin sec 2 sin œœ Ê œ ) ) 1 2 sin cos 1 sin 2 2 Ê œÊœ ) ) ) 11 # 4 r 2 sin 2 the point of intersection is œ Ê 1 4 È 2 . No other points of intersection exist. Š‹ È ß 1 4 54. r 2 csc and r 4 cos 2 csc 4 cos Ê ² ) ) 1 2 sin cos 1 2 , Ê œ ) ) ## 5 , ; r 4 cos 2 2 ; œ Ê œ² 1 1 44 4 4 5 È r 4 cos 2 2 . The point of ) œÊ œ ² œ 55 È intersection is 2 2 and the point 2 2 is the Š ÈÈ ß² ß 5 same point.
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Chapter 10 Practice Exercises 681 55. r cos 2 3 r cos cos sin sin ˆ‰ ˆ È )) ) ±œ Ê ² 11 1 33 3 2 3 r cos r sin 2 3 œÊ ² œ ÈÈ " ## È 3 r cos 3 r sin 4 3 x 3 y 4 3 ʲ œ Ê ² œ È È yx 4 Êœ ² È 3 3 56. r cos r cos cos sin sin ˆ ) ²œÊ ± 3 44 4 2 1 È # r cos r sin x y 1 ² ± ² ± œ È È 22 2 2 # # yx1 Êœ± 57. r 2 sec r r cos 2 x 2 œ Êœ Ê œ 2 cos ) 58. r 2 sec r cos 2 x 2 œ² Ê Ê œ² È 59. r csc r sin y Ê œ² Ê œ² 3 # 60. r 3 3 csc r sin 3 3 y 3 3 œ Ê œ È
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682 Chapter 10 Conic Sections and Polar Coordinates 61. r 4 sin r 4r sin x y 4y 0 œ± Ê Ê ² ² œ )) ## # x (y 2) 4; circle with center ( 2) and Ê ² ² œ !ß± radius 2. 62. r 3 3 sin r 3 3 r sin œÊ œ ÈÈ # xy3 3 y 0 x y ; ʲ ± œ ʲ± œ # # # È Š‹ 33 27 4 È circle with center and radius 63. r 2 2 cos r 2 2 r cos œ # xy2 2 x x 2 y2 ; ± œ Ê ± ² œ # # circle with center 2 0 and radius 2 ß 64. r 6 cos r 6r cos x y 6x 0 Ê Ê # (x 3) y 9; circle with center ( 3 0) and ʲ²œ ± ß radius 3 65. x y 5y 0 x y C # # ²²œÊ ²² œ Êœ ! ß ± ˆ‰ ˆ 52 5 5 4 and a ; r 5r sin 0 r 5 sin œ² œ Ê œ ± 5 # # 66. x y 2y 0 x (y 1) 1 C ( 1) and # # ²±œÊ ²± œÊœ ! ß a 1; r 2r sin 0 r 2 sin œ Ê œ #
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Chapter 10 Practice Exercises 683 67. x y 3x 0 x y C ## # # ±²œÊ ² ±œÊœß ! ˆ‰ ˆ 39 3 4 and a ; r 3r cos 0 r 3 cos œ² œ Ê œ 3 # # )) 68. x y 4x 0 (x 2) y 4 C ( 2 0) ±±œÊ ± ±œÊœ² ß and a 2; r 4r cos 0 r 4 cos œ± œ Ê œ ² # 69. r e 1 parabola with vertex at (1 0) œÊ œ Ê ß 2 1c o s ± ) 70. r r e ellipse; œ Ê œ Ê 84 2c o s 1 c o s ±# ± " ) ) " # ke 4 k 4 k 8; k ea 8 a œÊ œÊœ œ²Êœ ² "" aa e " # a ea ; therefore the center is Êœ Ê œ œ 16 16 8 33 3 ˆ‰ˆ ‰ " # ; vertices are ( ) and 0 88 ß) ß ß 11 71. r e 2 hyperbola; ke 6 2k 6 œ ÊœÊ œÊ œ 6 12 c o s ² ) k 3 vertices are (2 ) and (6 ) ß ß 72. r r e ; ke 4 œ Ê œ œ 12 4 3s i n 3 s i n ± ± " ) ) " 3 k 4 k 12; a 1 e 4 a 1 Êœ Ê œ ² œ Ê ² # # ab ’“ 4 a ea ; therefore the œÊœÊ œ œ 99 3 3 # " ˆ‰ˆ‰ center is ; vertices are 3 and 6 ˆ ˆ ‰ˆ‰ 3 # # ßß ß 1 73. e 2 and r cos 2 x 2 is directrix k 2; the conic is a hyperbola; r r œ œÊœ œ ) ke 1 e cos 1 cos (2)(2) ±± # r 4 c o s )
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684
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This note was uploaded on 10/13/2011 for the course MATHEMATIC 103 taught by Professor Thommas during the Spring '11 term at LCC Intl University.

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ISM_T11_C10_I - 680 Chapter 10 Conic Sections and Polar...

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