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ANSWERS
Mathematics 4 (Mathematical Analysis)
page 1
Chapter 10
Conic Sections: READINESS FOR COLLEGE MATHEMATICS TEXT
CIRCLES
PROBLEM SET A
4 )2 + ( y + 1)2 = 36
3.
3)2 + ( y - 1)2 = 25
5.
3) 2 + ( y - 8 ) 2 = 9
7.
3)2 + ( y + 2 )2 = 81
9. C(0, 5); r = 4
11. C(-3, 0); r = 6
- 5 - 1
13. C 2 , 2 ; r = 226
1.
(x (x (x (x -
(
)
2.
4.
6.
8.
10.
12.
14.
x 2 + ( y + 6 )2 = 32
( x + 1)2 + ( y - 5)2 = 26
( x - 5)2 + ( y + 1)2 = 49
( x - 5)2 + ( y + 3)2 = 25
C(3, -7); r = 2 5
C(1, -4); r = 5
C(-1, 3); r = 3 2 6
PROBLEM SET B
15.
17.
18.
19.
( x - 5)2 + ( y - 1)2 = 400
16. ( x - 2 )2 + ( y + 3)2 = 17
Two solutions: x2 + y2 = 25 and x2 + y2 = 225
(x - 4)2 + (y + 6)2 = 16; (x + 4)2 + (y - 18)2 = 16
( x - 5)2 + ( y - 5)2 = 25
PROBLEM SET C
5
, 5) or C(5 + 5 2, 5)
20. C( tan 22.5
21. (x - 6)2 + (y - 7)2 = 25
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22. x 2 ( 8p - 2 15 )
v.3.1
ANSWERS
Mathematics 4 (Mathematical Analysis)
page 2
ELLIPSES
PROBLEM SET A
1.
2.
3.
4.
5.
6.
7.
9.
11.
13.
C(0, 0), M = 20, m = 16 Foci (-6, 0) and (6, 0)
C(0, 0), M = 26, m = 10 Foci (0, -12) and (0, 12)
C(4, -1), M = 26, m = 24 Foci (4, 4) and (4, -6)
C(-3, 0), M = 10, m = 8 Foci (0, 0) and (-6, 0)
C(0, -5), M = 12, m = 4 5 Foci (4, -5) and (-4, -5)
C(4, 3), M = 10 2 , m = 2 Foci (-3, 3) and (11, 3)
C(0, 0), M = 10, m = 4
8. C(0, 0), M = 4, m = 2 2
C(-1, 3), M = 6, m = 2
10. C(0, -2), M = 10, m=4
12. C(-2, 3), M = 4 2 , m = 2 5
C(3, -5), M = 4 5, m = 6
2
x2 + y = 1
25 16
14.
2
x2 + y = 1
25 169
PROBLEM SET B
(x- 2)2 (y- 10)2
64 + 100 = 1
16.
15.
17. M = 20, m =16
18. a) A = 4x 16 - x 2
(x+3)2 (y- 5)2
=1
13 +
9
b) Max = 32
PROBLEM SET C
19.
(x- 3)2 (y- 9)2
9 + 81 = 1
or
(x+3)2 (y+9)2
9 + 81 = 1
20.
5+5 5
4
HYPERBOLAS
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v.3.1
ANSWERS
Mathematics 4 (Mathematical Analysis)
page 3
PROBLEM SET A
1.
y = ± 23 x, (± 13, 0)
2.
y = ± 45 x, (0, ± 41)
3.
y = ± 43 (x ± 3) ± 2; (3, 3), (3, - 7)
5.
y = ± 22 (x + 4) + 2; (- 4 ± 2 3, 2)
6.
7.
9.
y = ± 22 (x + 2) ± 3; (- 2, - 3 ± 30 )
C(-2, 5), t = 2, c = 4
8. C(-1, -4), t = 6, c = 10
C(-3, 0), t = 2 6 , c = 6
10. C(1, -3), t = 2 5, c = 4
2
4.
y = ± 15 (x + 3); (- 3 ± 26, 0)
2
y
11. x64 ± 36 = 1
2
(y±4)2
12. (x±2)
±
36
144 = 1
2
(y±5)2
13. (x+2)
±
9
16 = 1
14.
(y+2)2 (x±1)2
± 44 = 1
4
PROBLEM SET B
15.
(y+1)2 (x+3)2
(x+3)2 (y+1)2
=
1
or
16
36
81 36 = 1
17.
(
18.
2, 2 ) and
(-
2, -
2)
x2
y2
x2
y2
= 1 or
= 1 tan 22.5˚ can be replaced
16 tan 2 22.5
16 tan 2 67.5
by the value 2 - 1 and tan 67.5˚ can be replaced by 2 + 1.
PARABOLAS
PROBLEM SET A
1.
3.
5.
7.
V(0, 0); F(0, 3); y = -3
V(0, 0); F( - 18 , 0); x = 18
V(-3, 1); F(-3, 2); y = 0
13 , 2); x = 11
V(1, 2); F( 12
12
2
9. x = 12y
11. (y - 1)2 = -8(x - 3)
13. V(3, 1) and F(3, -1)
© 2001 CPM Educational Program
2.
4.
6.
8.
10.
12.
14.
V(0, 0); F(-2, 0); x = 2
1 ); y = - 1
V(0, 0); F(0, 16
16
V(-2, -5); F(2, -5) x = -6
41
V(1, 2); F(1, 39
20 ); y = 20
y2 = -20x
(x - 2)2 = 12(y - 3)
V(-1, 3) and x = 3
v.3.1
ANSWERS
Mathematics 4 (Mathematical Analysis)
page 4
PROBLEM SET B
16.
(y - 8)2 = 12(x - 4) or (y - 8)2 = -12(x - 10)
17.
(x + 4)2 = 8(y + 3)
PROBLEM SET C
19.
(x + 3)2 = 16(y + 1) or (x + 3)2 = -4(y - 4)
20.
(y - 2)2 = 16(x + 1)
21.
2
3
A = 4xc2c- x
SECOND-DEGREE EQUATIONS
PROBLEM SET A
2
1.
Parabola, x = 2(y+1)2
3.
5.
7.
Hyperbola, 4 ± (x+2)
=1
9
Point, (-3, 2)
Circle, (x + 12 ) 2 + (y ± 3)2 = 4
(y±1)2
2
(y±2)2
2.
4.
Ellipse, (x+1)
4 + 16 = 1
Parabola, -4(y + 1) = (x+2)2
6.
Intersecting lines, y = ±2(x + 3)
8.
2
(y+1)2
Hyperbola, (x+3)
±
=1
9
4
4
x2
9.
– 16y = 0
2 + 4y2 – 28x + 32 = 0
11. 3x
13. 2 3 2
x2
3y2
10.
–
– 40y – 48 = 0
2
12. y + 4x – 6y – 3 = 0
14. 46
PROBLEM SET B
15. x = ±6.25
16. x = 23
17. (x – 7)2 + (y – 3)2 = 25 where y π -2
18. 2x2 + 2xy + 2y2 + 12x + 12y – 36 = 0
19. 11x2 – 16xy – y2 + 36x + 32y – 436 = 0
PROBLEM SET C
20. x = 0.75
© 2001 CPM Educational Program
2
(y- 7)2
21. (x±7)
+
100 = 1
25
v.3.1