thomasET_226348_ism52

thomasET_226348_ism52 - Section 10.8 Conic Sections in...

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Section 10.8 Conic Sections in Polar Coordinates 667 41. r r r œÊ œ Ê œ 400 25 16 8 sin 1 s i n s i n ± ± ± ) ) ) ˆ‰ 400 16 8 16 " # e , k 50 y 50; a 1 e ke œœ Ê œ ± œ " # # ab a 1 25 a 25 a Ê ± œ ’“ " # # 3 100 43 ea Êœ 50 3 42. r r e 1, 43. r r e 1, œ Ê œ œ Ê œ 12 4 8 4 33 s i n 1s i n 22 s i n i n ±± ²² )) k4 y4 y 4 œÊœ ± 44. r r e , k 4 œ Ê œ œ 42 2s i n s i n ²# ² " ) ) " # y 4; a 1 e ke a 1 2 ± ± œ Ê ± œ # " # # a2 a e a Ê œ Ê œ 38 4 3 45. 46. 47. 48.
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668 Chapter 10 Conic Sections and Polar Coordinates 49. 50. 51. 52. 53. 54. 55. 56. 57. (a) Perihelion a ae a(1 e), Aphelion ea a a(1 e) œ ± œ± œ ² œ² (b) Planet Perihelion Aphelion Mercury 0.3075 AU 0.4667 AU Venus 0.7184 AU 0.7282 AU Earth 0.9833 AU 1.0167 AU Mars 1.3817 AU 1.6663 AU Jupiter 4.9512 AU 5.4548 AU Saturn 9.0210 AU 10.0570 AU Uranus 18.2977 AU 20.0623 AU Neptune 29.8135 AU 30.3065 AU Pluto 29.6549 AU 49.2251 AU
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Section 10.8 Conic Sections in Polar Coordinates 669 58. Mercury: r œœ (0.3871) 1 0.2056 1 0.2056 cos 1 0.2056 cos 0.3707 ab ± ²² # )) Venus: r (0.7233) 1 0.0068 1 0.0068 cos 1 0.0068 cos 0.7233 ± # Earth: r 1 1 0.0167 1 0.0167 cos 1 0.0617 cos 0.9997 ± # Mars: r (1.524) 1 0.0934 1 0.0934 cos 1 0.0934 cos 1.511 ± # Jupiter: r (5.203) 1 0.0484 1 0.0484 cos 1 0.0484 cos 5.191 ± # Saturn: r (9.539) 1 0.0543 1 0.0543 cos 1 0.0543 cos 9.511 ± # Uranus: r (19.18) 1 0.0460 1 0.0460 cos 1 0.0460 cos 19.14 ± # Neptune: r (30.06) 1 0.0082 1 0.0082 cos 1 0.0082 cos 30.06 ± # 59. (a) r 4 sin r 4r sin x y 4y; œÊ œ Ê ± œ ## # r 3 sec r r cos 3 œ Ê œ ÈÈ È 3 cos ) x3 ; 3y 4 y Êœ œ Ê ±œ ÈÈÈ Š‹ # # y 4y 3 0 (y 3)(y 1) 0 y 3 ʲ ± œ ² œ Ê œ # or y 1. Therefore in Cartesian coordinates, the points œ of intersection are 3 3 and 3 1 . In polar ßß coordinates, 4 sin 3 sec 4 sin cos 3 ) ) œ 2 sin cos sin 2 2 or Ê œ Ê œ ) ) 33 3 1 or ; r 2, and 2 36 3 6 3 11 1 1 1 œÊœ œ ) r 2 3 2 and 2 3 are the points Ê ß ß ˆ‰ 63 of intersection in polar coordinates. (b) 60. (a) r 8 cos r 8r cos x y 8x œ Ê ± œ # x 8x y 0 (x 4) y 16; Ê ²±œÊ² ±œ # # r 2 sec r r cos 2 œ 2 cos ) x2 ; 2 8 ( 2 )y 0 Êœ œÊ ² ±œ y 12 y 2 3. Therefore 2 2 3 ÊœÊ œ ß # are the points of intersection in Cartesian coordinates. In polar coordinates, 8 cos 2 sec 8 cos 2 ) œ # cos cos , , , or Ê œ Ê œ # "" # ) 43 3 3 24 111 ; and r 4, and and 55 2 4 3 3 3 1 1 1 œ œ r 4 4 and 4 are the points of ²Ê ß ß ˆ ‰ˆ‰ 5 (b) intersection in polar coordinates. The points 4 and 4 are the same points. ²ß 61. r cos 4 x 4 k 4: parabola e 1 r ) œÊœÊœ ʜʜ 4 1c o s ² ) 62. r cos 2 r cos cos sin sin 2 r sin 2 y 2 k 2: parabola e 1 ˆ ² œÊ ± 1 # r 2 1s i n ² )
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670 Chapter 10 Conic Sections and Polar Coordinates 63. (a) Let the ellipse be the orbit, with the Sun at one focus. Then r a c and r a c max min rr œ± œ² Ê max min max min ± ² e œœ œ œ (a c) (a c) (a c) (a c) 2a a 2c c ²±± ²²± (b) Let F , F be the foci. Then PF PF 10 where "# " # ±œ P is any point on the ellipse. If P is a vertex, then PF a c and PF a c œ² (a c) (a c) 10 Ê ±±²œ 2a 10 a 5. Since e we have 0.2 ÊœÊ œ œ œ cc a5 c 1.0 the pins should be 2 inches apart. Êœ Ê 64. e 0.97, Major axis 36.18 AU a 18.09, Minor axis 9.12 AU b 4.56 (1 AU 1.49 10 km) Ê œ œ Ê œ ¸ ) (a) r AU œœœ œ ke 1.07 1e c o s c o s 10 . 9 7 c o s . 9 7 c o s a1 e (18.09) 1 (0.97) ²² ² ² ± ± )) ) ) ab cd # # (b) 0 r 0.5431 AU 8.09 10 km ) œÊœ ¸ ¸ ‚ 1.07 . 9 7 ² ( (c) r 35.7 AU 5.32 )1 ¸ 1.07 . 9 7 ± * 65. x y 2ay 0 (r cos ) (r sin ) 2ar sin 0 ## # # ±² œÊ ± ² œ ) r cos r sin 2ar sin 0 r
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thomasET_226348_ism52 - Section 10.8 Conic Sections in...

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