4-3

/ *

4-3

* /

F1 (3,0), F2 (3,0) , Sk(x-3)2+y2=k2, Tk(x+3)2+y2=k2, k=1,2,,7,

4-144-14p.2271-01-11-21-3

* /

p.2271-01-11-21-3

kSkTk+4Sk+4TkPk, QkPk+4, Qk+4

( )F1(3, 0)F2(3,0)k=1P1P5155144k=2P2, Q2P6, Q6266244k=3P3, Q3P7, Q7377344k=4P4, Q4P8, Q8488444

* /

, P1, P2, P3 , , P8 , Q2, Q3 , , Q8F1 (3 , 0)F2 ( 3 , 0)4. ,4-14, . ,

p.2271-01-11-21-3

* /

F1F2, a0, 2a, | |2a P, F1F2.p.2281-01-11-21-3

* /

, . F1F2 . , 2a, 2a . F1, F2 . , , , P, .

p.2281-01-11-21-3

* /

, P , , 4-15 , P 2a. , , 4-15, P 2a. .p.2281-01-11-21-3

* /

p.2281-01-11-21-3

* /

1-0 6, | |4P , , (1) 4P (2) 4P a0,(1) 2a, P| |2a, P (2) 2a, P| |2a, P p.2291-01-11-21-3

* /

1-0(1). (2). 6, | |4P , , (1) 4P (2) 4P p.2291-01-11-21-3

* /

1-0(1). (2). a0,(1) 2a, P| |2a, P (2) 2a, P| |2a, P p.2291-01-11-21-3

* /

, , . F1, F2, . . , . . . .4-16p.2291-01-11-21-3

* /

1-01-11-21-3

* /

F1, F2, , P, | |2a, 2a.

A,B, p.2291-01-11-21-3

/*

, 2a.p.2301-01-11-21-3

* /

1-1 O , = = , ., .p.2301-01-11-21-3

* /

2a, 2c, ,OC, D, b,b , .

.4-17p.2301-01-11-21-3

/*

, PF1F2Q,F1F2 , , , | |2a,

| |2a,

Q, F1F2,. .p.2301-01-11-21-3

* /

1-2PCDQ,CD .P, QAB, ABS, R,, PQRS, = , = , = ,PSF1 =QRF2 = 90,PSF1 QRF2, = , = ..p.2301-01-11-21-3

* /

1-2 = 2a, = 2a,Q,.P OR, = , = ,PF1RF2, = , = , = 2a, = 2a,.p.2301-01-11-21-3

* /

1-2R, O..p.2301-01-11-21-3

* /

F1,,F1F2, . x, F1, F2, 2c, 2a, 2b,

p.2311-01-11-21-3

/*

2a, 2ax.PF1F2 , (2a+x)2=(2c)2+x2, axc2a2,b , b2c2 a2, axb2, x p.2311-01-11-21-3

* /

1-310, 8,2c = 10, 2a = 8, c = 5, a = 4,c2 = a2 + b2b2 = 52 42 = 9,

10, 8, .p.2311-01-11-21-3

/ *

, , ,.p.231 1

* /

1| |4| |4, 4, 4,x26x9y2(x26x9)y28 16,F1(3, 0), F2(3, 0),P(x, y), | |4, P.p.232 1

* /

1 3x4 2 ,9x224x164[(x26x9)y2], 5x24y220,PF1(3, 0), F2(3, 0),P(x, y), | |4, P.p.232 1

* /

.1. F1(c, 0), F2(c, 0), c>0, x , y, , 4-18. P(x, y), | |2a, ca0, 2a, 2a,4-18p.23222334-14

* /

2a,x22cxc2y2x22cxc2y24a 4a2, cxa2 a , a42a2cxc2x2a2x22a2cxa2c2a2y2,(c2a2)x2a2y2a2(c2a2), b , b2x2a2y2a2b2,Pp.23222334-14

* /

, P(x,y)b , 211 , P(x, y).| |2ap.23322334-14

* /

2.F1(0,c), F2(0,c) ', c>0, y, x, , ' , p.23322334-14

* /

ca0, b , F1(c,0), F2(c,0),| |2aPp.23322334-14

* /

ca0, b , F1(0,c), F2(0,c),| |2aPa, b, cc2a2b2.p.23322334-14

* /

2 (13,0), c13, x, . 24 , 2a24 , a12 .. , (13, 0), 24. (0, 5)(0, 5), (0, 3).p.23422334-14

* /

2c2a2b2 , b2c2a216914425,. , (13, 0), 24. (0, 5)(0, 5), (0, 3).p.23422334-14

* /

2 (0, 5)(0, 5), , c5, y, . (0, 3), a3, c2a2b2 b2c2a225916,. , (13, 0), 24. (0, 5)(0, 5), (0, 3).p.23422334-14

* /

2. , (13, 0), 24. (0, 5)(0, 5), (0, 3).p.23422334-14

* /

2(3, 0)( 3, 0),10,, a = 3, c = 5. = 25 9 = 16,x,x, .(3, 0)( 3, 0), 10., (0, 6), 10.p.23422334-14

* /

2, (0, 6),c = 6.10, 2a = 10, a = 5, = 36 25 = 11,y,y, .(3, 0)( 3, 0), 10., (0, 6), 10.p.23422334-14

* /

3

(0, 0), a5, b3, .c2a2b2 c .

. .p.23522334-14

* /

3 (0, 0), (5, 0) (5, 0), ( ,0) ( ,0). 2a10, 2b6,

. .p.23522334-14

* /

3

a = 3, b = 6, c = = 3 .

..p.23522334-14

* /

3(0, 0),(0, 3)(0, 3),(0, 3 )(0, 3 ).2a = 6,2b = 12,

= 24.

..p.23522334-14

* /

, LORAN. , , , , , , , , p.23522334-14

* /

4A , B , 200 , A B . A , B , , AB120 , AB120 . ,A , B , AB x , A , B , .p.2362234-143

* /

4,c100. 120,,2a, a60.c2a2b2 b2c2a210026026400,x, p.2362234-143

* /

4-140, y = 40x.

B, ( , 40).4, 40, .p.2362234-143

* /

Pyf( x ), P yf( x ), PL0, 0(PL, ), Lyf(x).

y=2x , x, 2x0 , 0 , y=2xx , x , x y=2x. 4-19

p.2364-24-3

* /

y=log2xx>0, x0, y=log2xy , y , y y=log2x. 4-204-194-20p.2374-24-3

* /

.

L

L' P(x0, y0), PLL'.p.2374-24-3

/*

LL 'Lbx ay=0, L ' bx+ay=0.P(x0, y0) , b2x02a2y02a2b2,(bx0ay0)(bx0ay0)a2b20,PLL'. p.2374-24-3

/*

PLL'

d d ' dd ' k PL L'.p.2374-24-3

* /

, P,PL'd '

,P L d= 0 , L

.

L'

. p.2384-24-3

* /

.

' p.2384-24-3

* /

4-2 4x 5y = 04x + 5y = 0.

.

p.2384-24-3

* /

A1(a, 0), A2(a, 0), B1(0, b), B2(0, b).A1,A2xxaxa, B1,B2yybyb,, 4-21p.2384-24-3

* /

Ly x L ' y x,

L L '

LL'., ,.p.2384-24-3

* /

a = b, 2a2b, . 4-21 , , . ' ' ,p.2394-24-3

* /

, '. , , 4-22.4-22p.2394-24-3

* /

4-3

, 3x 4y = 0 3x + 4y = 0.

.

, .p.2394-24-3

* /

4-3 , 3x 4y = 03x + 4y = 0.

.

, .p.2394-24-3

* /

(a0, b0), (h, k), '

, '(h, k),(hc, k)(hc, k). ,p.239576657

* /

(h, k), ' , '(h, k),(h, kc)(h, kc). .p.239576657

* /

, , , a0, b0, c= .p.240576657

* /

p.240576657

1 1 (h, k)(h, k)(ha, k), (ha, k)(h, ka),(h, ka)(hc, k) ,(hc, k)(h, kc) ,(h, kc)yk, xhxh, yk

* /

p.240(h, k)(0, 0), .576657

2a2a2b2b 0 0

* /

5

(2, 3),y3, a4, b3,.p.241576657

* /

5 c2a2b2 (2, 3), (24, 3)(6, 3), (24, 3)(2, 3). (25, 3)(7, 3), (25, 3)(3, 3).p.241576657

* /

52a8, 2b6,p.241576657

* /

5 3x4y1803x4y60.

p.241576657

* /

5

(1, 2),x = 1,a = 5, b = 3, ,

... p.241576657

* /

5

(1, 2),(1, 2 5) = (1, 7), (1, 2 + 5) = (1, 3).

... p.241576657

* /

52a = 10,2b = 6,

... ) p.241576657

* /

5 5x 3y 11 = 05x + 3y + 1 = 0

... p.241576657

* /

6F1(6,1)F2(4,1), C(1, 1), x, .A(4, 1),. F1(6, 1)F2(4, 1),A(4, 1). (1, 2), F1(1,8),10.p.242576657

* /

6 c2a2b2 b2c2a225916,

. F1(6, 1)F2(4, 1),A(4, 1). (1, 2), F1(1,8),10.p.242576657

* /

6 C(1, 2), F1 (1, 8), c 2(8)6, y, . 2a10, a5,

. F1(6, 1)F2(4, 1),A(4, 1). (1, 2), F1(1,8),10.p.242576657

* /

6 c2a2b2 b2c2a2362511, . F1(6, 1)F2(4, 1),A(4, 1). (1, 2), F1(1,8),10.p.242576657

* /

6F1(1, 2), F2(7, 2),C(3, 2),c = = | 1 (3) | = 4.

c2 = a2 + b2a2 = 16 7 = 9,.F1(1, 2)F2(7, 2), (1, 1)(1, 3), F1(1, 3).p.242576657

* /

6 .F1(1, 2)F2(7, 2), (1, 1)(1, 3), F1(1, 3).p.242576657

* /

6(1, 1), (1, 3), C(1, 1), a = | 3 1 | = 2.F1(1, 3), c = | 1 (3) | = 4,c2 = a2 + b2 b2= 16 4 = 12,.F1(1, 2)F2(7, 2), (1, 1)(1, 3), F1(1, 3).p.242576657

* /

6

.F1(1, 2)F2(7, 2), (1, 1)(1, 3), F1(1, 3).p.242576657

* /

5, 9(x24x4)16(y26y9)169, 9x216y236x96y2520,.,

p.243576657

* /

Ax2Cy2DxEyF0, A0, C0.

, Ax2Cy2+Dx+Ey+F=0AC > 0, .

p.243576657

* /

7 9(x24x22)25(y22y12)2143625, 9(x2)225(y1)2225 , .. 9x225y236x50y2140. 9x225y236x50y110.p.243576657

* /

7 9(x24x22)25(y22y12)113625, 9(x2)225(y1)20 , [3(x2)5(y1)][3(x2)5(y1)]0, (3x5y11)(3x5y1)0, 3x5y1103x5y10.. 9x225y236x50y2140. 9x225y236x50y110.p.243576657

* /

79(x2 + 2x + 12) 4(y2 4y + 22) = 29 + 9 16,9(x +1)2 4(y 2)2 = 36, 9x2 4y2 + 18x + 16y + 29 = 0p.243576657

* /

4-2, .p.24488-18-2

* /

8

Q(x', y '),

A. 'p.24488-18-2

* /

8

A. p.24488-18-2

* /

8

A. p.24488-18-2

* /

8-1 A',

A. p.24488-18-2

* /

8-1

A.p.24488-18-2

* /

8-1 , '

A.p.24488-18-2

* /

8

. rs, . Q(x', y '), P(x, y),p.24588-18-2

* /

p.24588-18-2

* /

, (0, 0), ra, rb,

p.24588-18-2

* /

r.

p.24588-18-2

* /

8-22.

p.24588-18-2

* /

1. 2.3.

, 1.P(x, y) P.2..3..p.246

* /

1. P.2.., 1.P(x, y) P.2..3..p.246

* /

4.5.4.(1, 2), 10, 6, 5. .p.246

* /

4. 5.. 4.(1, 2), 10, 6, 5. .p.246

* /

1.., (5, 0), 8.(0, 17)(0, 17), (0, 15).(8, 1)(2, 1),(5, 1).(1, 2), (1, 10), 12.x = 1, y = 3, 4, p.246

* /

(5, 0), c = 5.8,2a = 8, a = 4.

1.., (5, 0), 8.p.246

* /

(0, 17), (0, 17), , c = 17,(0, 15), a = 15, = 289 225 = 64,

1..(0, 17)(0, 17), (0, 15).p.246

* /

(8, 1), (2, 1), (3, 1), c = | 8 3 | = 5,(5, 1), a = | 5 3 | = 2,

1..(8, 1)(2, 1),(5, 1).p.246

* /

(1, 2), (1, 10), c = | 10 2 | = 8,12, 2b = 12, b = 6,

1..(1, 2), (1, 10), 12.p.246

* /

x = 1, y = 3, (1, 3). 4, 2c = 4, 2a =1..x = 1, y = 3, 4, p.246

* /

1..x = 1, y = 3, 4, p.246

* /

a = 3, b = 4,c2 = a2 + b2c2....p.246

* /

(1, 1),(2, 1)(4, 1), ( 4, 1)(6, 1).2....p.246

* /

2a = 6,2b = 8, 2....p.246

* /

4x 3y 7 = 04x + 3y 1 = 0.

2....p.246

* /

y, a > 0, b > 0.

2a = 10, 2c = 26, a = 5, c = 13.3.F1(0, 13)F2(0, 13), P, .p.246

* /

c2 = a2 + b2b2 = 132 52 = 144,5x 12y = 05x + 12y = 0.

5x 12y = 05x + 12y = 0.3.F1(0, 13)F2(0, 13), P, .p.246

* /

(3, 0), (3, 0),, x, (5, 4),

4.(3, 0)(3, 0), (5, 4), .p.247

* /

b2= 9 a2, 25( 9 a2) 16 a2 = a2( 9 a2),a4 50a2 + 225 = 0, (a2 5)(a2 45) = 0a2= 545a = , b2 = 9 5 = 4,

4.(3, 0)(3, 0), (5, 4), .p.247

* /

a = 3, b = 2,a2 = b2 + c2 c2 = 9 4 = 5, ( , 0)( , 0), (0, 0),,

5. , .p.247

* /

a2 + b2 = 5, 2a = 4, a = 2, b2 = 5 4 = 1,5. , .p.247

* /

1 2 3 2 = 1, 3 = 4 1, 3 = 2 = 4 1.6.1 , 2 , 3 1, 2, 3, 1, 2, 3.p.247

* /

, x y = 01, 1,(1, 1), y 1 = (x 1), x + y 2 = 0.1.x y = 0, (1, 1), (5, 1), ...p.247

* /

(1, 1), (5, 1), c = | 5 1 | = 4. 1.x y = 0, (1, 1), (5, 1), ...p.247

* /

1.x y = 0, (1, 1), (5, 1), ...p.247

* /

c = 4, a =2 (1, 1),( )( ).(5, 1)(3, 1).

1.x y = 0, (1, 1), (5, 1), ...p.247

* /

(1,2)(1,4), (1, 1), a = | 4 1 | = 3. , a = 3, b = 4,2.(1,2)(1,4), , .p.247

* /

2.(1,2)(1,4), , .p.247

* /

3. , , .p.247

* /

a = 3, b = 4. c2 = 9 + 16 = 25, c = 5.4.P , F1,F2 , , p.247

* /

= t, = 3t, t0, | | = | 3t t | = 2t = 6,t = 3, = + + = 3 + 9 + 10 = 22.4.P , F1,F2 , , p.247