Thomas Calculus 13th [Solutions]

CHAPTER 11 ADDITIONAL AND ADVANCED EXERCISES
Chapter 11 Additional and Advanced Exercises 871
1. Directrix x 3 and focus (4, 0) vertex is 7
2 , 0
p 1 the equation is
2
x
2
7 y
2 2
2.
2
2 2
( x 3)
x 6x 12y 9 0 x 6x 9 12 y y vertex is (3, 0) and p 3 focus is (3, 3) and
12
the directrix is y 3
3.
2
x 4y vertex is (0, 0) and p 1 focus is (0, 1); thus the distance from P( x, y ) to the vertex is
2 2
x y and the distance from P to the focus is
2 2 2 2 2 2
x ( y 1) x y 2 x ( y 1)
2 2 2 2 2 2 2 2 2 2
x y 4 x ( y 1) x y 4x 4y 8y 4 3x 3y 8y 4 0, which is a circle
4. Let the segment a b intersect the y -axis in point A and
intersect the x -axis in point B so that PB b and
PA a (see figure). Draw the horizontal line through P
and let it intersect the y -axis in point C. Let PBO
x
a
APC . Then sin
2 2
y
2 2
b
y
b
2 2
cos sin 1.
and cos x
a
5. Vertices are
(0, 2) a 2; e c 0.5 c c 1 foci are (0, 1)
a 2
6. Let the center of the ellipse be ( x , 0); directrix x 2, focus (4, 0), and 2 a 2 a
3 e
e
2 c
a 2 (2 ).
3 c ae 2 a a 2 2 2 a a 4 4 a 5 a 4 a 12 ;
3 3 3 3 9 9 3 5
x 2 a x 2 12 3 18 x 28 the center is 28 , 0 ; x 4 c c 28 4 8 so that
e
5 2 5 5
5 5 5
2 2 2 28
2
2
c a b 2
8
2
80
5 5 25 ; x
2
5 y 25 x
28 2
5 5y
1 or
144 80
144 16
1
25 25
7. Let the center of the hyperbola be (0, y).
(a) Directrix y 1, focus (0, 7) and 2 a 6 a
e e
c 6 a 2 12. c ae 2 a
a 2(2 a) 12 a 4 c 8; y ( 1) a 4
e 2
2 y 1 the center is (0,1);
2 2 2 2 2 2
( y 1)
c a b b c a 64 16 48; therefore the equation is
2
x
16 48
1
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