Thomas Calculus 13th [Solutions]

870 Chapter 11 Parametric Equations and Polar Coordinates
81. e 2 and r cos 2 x 2 is directrix k 2; the conic is a hyperbola;
r
4
1 2cos
82. e 1 and r cos 4 x 4 is directrix k 4; the conic is a parabola;
r
4
1 cos
r
r
ke
(2)(2)
r
1 ecos 1 2cos
ke
(4)(1)
r
1 ecos 1 cos
83. e 1 and r sin 2 y 2 is directrix k 2; the conic is an ellipse;
2
r
2
2 sin
r
1
2
1
2
(2)
ke r
1 esin 1 sin
84. e 1 and r sin 6 y 6 is directrix k 6; the conic is an ellipse;
3
r
6
3 sin
r
1
3
1
3
(6)
ke r
1 esin 1 sin
2 2 2 2 2
85. (a) Around the x-axis:
9x 4y 36 y 9 9 x y 9 9 x and we use the positive root:
4 4
2
2
2 2 2 3
2
V 2 9 9 3
0 9 x dx
4 2 0 9 x dx
4 2 9 x x
4
24 0
2 2 2 2 2
(b) Around the y-axis:
9x 4y 36 x 4 4 y x 4 4 y and we use the positive root:
9 9
3
2
2 3 2 3
3
V 2 4 4 4
0 4 y dy
9 2 0 4 y dy
9 2 4 y y
27
16 0
86.
2 2 2 9 36 3 2
9x 4y 36, x 4 y x y x 4;
4 2
3 4
2
9 x 4x
9 64 16 8 8 9 56 24 3 (32) 24
4 3
2
4 3 3 4 3 3 4
4
2
3 2 9
4 2
V x
2 2 4 dx x
4 2
4 dx
87. (a)
(b)
2 2 2 2 2 2 2 2 2
r k r er cos k x y ex k x y k ex x y k 2kex e x
1 ecos
2 2 2 2 2 2 2 2 2
x e x y 2kex k 0 1 e x y 2kex k 0
2 2 2 2 2 2
e 0 x y k 0 x y k circle;
2 2 2 2 2 2
0 e 1 e 1 e 1 0 B 4AC 0 4 1 e (1) 4 e 1 0 ellipse;
2 2
e 1 B 4AC 0 4(0)(1) 0 parabola;
2 2 2 2 2
e 1 e 1 B 4AC 0 4 1 e (1) 4e 4 0 hyperbola
88. Let r 1,
1 be a point on the graph where r 1 a 1 . Let r 2,
2 be on the graph where r 2 a 2 and
2 1 2 . Then r 1 and r 2 lie on the same ray on consecutive turns of the spiral and the distance between
the two points is r 2 r 1 a 2 a 1 a 2 1 2 a , which is constant.
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