. , , . () , O . , - .
M OM (-) M. (.1.1). M (, y), (, ). .
. 1.1.
d M (i, yi) N (xj, yj)
(1.1)
d 2 = (x i- x j)2 + (y i - y j)2
. d M (-3,4) N ((5.2). (1.1)
.
. O, , OP, . . (. 1.2) M r (.. OM, M) j, ( ). j : -p<j p 0 j<2p. r j (r, j). r = 0.
, (.1.3),
= r cosj y = r sinj. (1.2)
2 + y 2 = r 2(cos2j+ sin2 j) = r 2 . (1.3)
tgj = . (1.4)
(1.2) M , (1.3) (1.4) .
. 1.2. . 1.3.
.
. 1.4. .
. X 1 Y 1 X 2 Y 2. M (x 1, y 1) (x 2, y 2) .
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() (. 1.4.). M (x 1, y 1), O 1 (0,0), O 2 - (,b), M (x 2, y 2). .
(1.5)
. OX 1 OX 2 j. . 1.5
(1.6)
.1.5. .
(1.7)
. M (-2,3), 300 ?
(1.7) x 1= -2, y1 = 3, j = 300, =0 b = 2,
x 2 = -2cos300 + 3sin300 = -2 + 3 = -
y 2 = 2sin300 +3cos300 - 2 = 2 + 3 -2 = - 1
B (0, b) j (.4.6). tg(φ) k.
M (x,y) ( ). .
tg j = = k (1.8)
. (1.8) y, " "
(1.9)
. 1.8. .
b = 0, . k = 0, = b. N (x 0, y 0) (. 1.9),
k = tg j = (1.10)
(1.10) " , ".
. 1.9. .
N (x 0, y 0) M (x 1, y 1), ,
(1.11)
" , "
. 1.11.
α k 1 =tg(φ1) k 2= tg(φ2) ( 1.11). , , , α = φ2 φ1.
(1.12)
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φ2= φ1: k 1= k 2;
φ2 φ1 = 900: 1= - k1k2 .
.
+ B y + = 0. (1.13)
, (1.9) A = k, B = -1, C = b. (4.10) (4.11). . + B y + = 0. 0, (1.13) (1.9)
, k = , b = .
= 0, .
= 0, 0, = (. 1.12). .
= 0, 0, + = 0 x = - . = ( = - ) (. 1.12.).
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.1.12. ,
+ + = 0 . : .
. ( 1, 1) ( 2, 2), ( 3, 3)
(1.14)
. (-3,-3), (2,7) (5,1) (. 1.13). , , .
. , , (1.11):
A : = 2 + 3. k = 2.
, (1.11)
: = 0,5 -1,5. k = 0,.5.
: = -2 + 11. k = -2.
. 1.13.
(1.12) :
, k1=2, k2 = 0,5.
. , , (1.14), ,
, . . , (1.10) :
, : - 0,5 +1,5 =0
,
Ax 2 + Bxy + C y2 + Dx + Ey + F = 0, (1.15)
, B, .
, . R M (, b)
(x - a)2 + (y - b)2 = R 2. (1.16)
a = b =0, . x 2 + y 2 = R 2. , , (1.16) (1.15),
A = C = 1, B =0, D =-2 a, E = -2 b, F = - R 2 + a 2 + b 2.
. 2 + y 2 - 4 x = 0. , , ?
. (1.16). , 4
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x 2 + y 2 - 4 x = (x 2 - 4 x + 4) + y 2 - 4 = 0
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(x - 2)2 + y 2 = 22. (1.17)
(1.16) (1.17), , R =2 M (2,0).
- , F 1 F 2, , (.. ) , , 2 ( > 0).
, F 1 (,0) F 2 (- ,0) (.1.14), . . MF 1 MF 2 , , 2 .
MF 1 + MF 2 = 2 ,
,
,
.
.4.14. . MF 1 + MF 2 = NF 1 + NF 2 = 2 ,
, ,
. (1.18)
a b ( 2 2 b , a,b, ),
a 2 = b 2+ c 2.
, < 1.
(1.18) (1.16)
B = D = E = 0, , F = -1.
, - , a = b = R, .
, , , 2 (. 1.15). , F 1 (,0) F 2 (- ,0) (.1.15), . . MF 1 MF2 , , 2 ( > 0).
MF 2 MF 1 = 2 .
, . ,
.
.
. 1.15. . MF2 MF1 = NF1 NF2 = 2.
,
. (1.19)
b ( a,b, ), , ,
c 2 = b 2 + a 2.
e> 1.
(1.19) (1.15)
B = D = E = 0, , F = - 1.
- .
.
- , (. 1.16) , , , . F F (,0), , , . (x, y) . . MK = MF.
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,
,
y 2 = 2 px. (1.20)
, . (1.20) (1.15)
A = B = E = F = 0, C = 1, D = - 2 p.
, (1.15) : (1.18), (1.19), (1.20) , , . , ( ), .
. 1.16. . NF = NL MK = MF
( ) ( ) (x 0, y 0) ( ).