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1 1 3 < x < 5; (b) x . 3 2 4 356
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1.19 Yes: 1.20 No: 1.21
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(a2 )2 = a4 and a2 0. a2 < b2 implies |a| < |b|, but a < b does not hold when, for example, a = 1 and b = 2.
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8 9 3 5 2 4 5 1 , BC = , IB + BD = 2 + = , ID = , IB + BC = 2 + = , IC = . 3 3 3 2 2 2 2 2
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1.22 (a) b = 10; (b) b = 5; (c) b = 5.
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2.4 A(0, 4), B(2, 2), C(4, 0), D( 3, 1), E(0, 4), F(2, 3). 2.6 (a) 5; (b) 5; (c) 2; (d) 8. 2.7 Area (right triangle) = 2.8 (3, 4). 1 1 (AC)(AB) = (5)(10) = 25. 2 2 2.10 (0, 2), (6, 2), (4, 4).
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2.9 ( 1, 1) and (3, 0).
2.11 (2, y) for some real number y. 2.12 (a) 34; (b) 3 2; (c) 389 . 4
2.13 (a) Isosceles only; (b) right only, area = 10; (c) isosceles right, area = 17. 2.14 k = 5. 2.15 (a) No; (b) yes. 5 5+ 2 , 2 ; (c) ,2 . 4 2 2.17 (5, 8).
2.16 (a) (4, 2); (b)
2.19 We may assume that A is the origin (0, 0) and that B = (b, 0) is on the positive x-axis. Let C be (x, y) and let D be (u, v). u v b b+x y x+u y+v Then M1 = , 0 , M2 = , , M3 = , , and M4 = , are the midpoints of AB, BC, CD, DA, 2 2 2 2 2 2 2 1 y+v 1 b x+u respectively. The opposite sides of ABCD are AB and CD. The midpoint of M1 M3 is + , = 2 2 2 2 2 u 1 y v b+x+u y+v b+x+u y+v 1 b+x , . The midpoint of M2 M4 is + , + = , . Hence, M1 M3 4 4 2 2 2 2 2 2 4 4 and M2 M4 bisect each other.
3.9 See Fig. A-1. 3.10 See Fig. A-2. 3.11 See Fig. A-3. 3.12 See Fig. A-4. 3.13 x 2 = 4py (parabola).
3.14 (a) (x 4)2 + ( y 3)2 = 1; (b) (x + 1)2 + ( y 5)2 = 2; (c) x 2 + ( y 2)2 = 16; (d) (x 3)2 + ( y 3)2 = 18; (e) (x 4)2 + ( y + 1)2 = 20; ( f ) (x 1)2 + ( y 2)2 = 5.
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Fig. A-1
Fig. A-2
3.15 (a) Circle [center (6, 10), radius 11]; (b) circle [center (0, 15), radius 14]; (c) null set; 1 1 , 0 , radius ; (e) point ( 1, 1); ( f ) circle [center ( 3, 2), radius 7]. (d) circle center 4 4 3.16 (b) 4F < D2 + E 2 . 3.17 (a) (x 3)2 + ( y + 2)2 = 100; (b) (x 4)2 + y2 = 26.
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Fig. A-3
4 3.18 k = 2 and k = . 5 3.19 (x 1)2 + ( y 3)2 = 9, (x 7)2 + ( y 3)2 = 9, (x 1)2 + ( y 9)2 = 9, (x 7)2 + ( y 9)2 = 9. 3.21 x+ 2 2 41 2 1300 + y = 7 7 49
3.22 x 2 + ( y 1)2 = 10
4.8 (a) y 5 = 1 1 (x 2) or y 4 = (x + 1); (b) y = 4x or y 4 = 4(x 1); 3 3 (c) y + 1 = (x 7) or y 7 = (x + 1). 5 14 2 x + ; (b) y = 2x 1; (c) y = x + 2; (d) y = 4; (e) y = 5x 1; ( f ) = 3x + 16; 6 3 3 13 2 3 (g) y = x + ; (h) y = 0; (i) y = 3x + 7; ( j) y = x; (k) y = 3; (l) y = x. 4 4 3
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