2 + 2 + 2 + 2Dx + 2Ey + F = 0.
( ), , .
1) - .
2) - .
3) - .
4) a2x2 c2y2 = 0 .
5) y2 = 2px .
6) y2 a2 = 0 .
7) y2 + a2 = 0 .
8) y2 = 0 .
9) (x a)2 + (y b)2 = R2 .
.
(x a)2 + (y b)2 = R2 (a; b).
. , :
2x2 + 2y2 8x + 5y 4 = 0.
, .9. :
x2 + y2 4x + 2,5y 2 = 0
x2 4x + 4 4 + y2 + 2,5y + 25/16 25/16 2 = 0
(x 2)2 + (y + 5/4)2 25/16 6 = 0
(x 2)2 + (y + 5/4)2 = 121/16
(2; -5/4); R = 11/4.
.
. , .
. , .
r1
r2
F1 O F2
F1, F2 . F1 = (c; 0); F2(-c; 0)
;
a ;
b .
. :
a 2 = b 2 + c 2.
: , , r 1 + r 2 = 2 ( ). , , r 1 + r 2 = a c + a + c. .. r 1 + r 2 , , , :
a 2 = b 2 + c 2
r 1 + r 2 = 2 a.
. , .
= /a.
.. < a, < 1.
. k = b/a , 1 k = (a b)/a .
: k2 = 1 e2.
|
|
a = b (c = 0, e = 0, ), .
(1, 1) : , , , .
. (, ), :
r1 = a ex, r2 = a + ex.
. , r1 + r2 = 2a. , :
:
, r2 = a + ex. .
, . :
x = a/e; x = -a/e.
. , , , .
. , , :
1) : x = 0; y2 = 16; y = -4.
2) : c2 = a2 b2 = 25 16 = 9; c = 3; F2(-3; 0).
3) , :
. , F1(0; 0), F2(1; 1), 2.
: . :
2c = , , a2 b2 = c2 = ½
2 = 2, = 1, b =
: .
.
. , , , .
y
M(x, y)
b
r1
r2
x
F1 a F2
c
ïr1 r2ï= 2a. F1, F2 . F1F2 = 2c.
(, ). :
2 2 = b2 ( )
.
, .
2 .
2b .
,
. , , .
, 2 2 = b2:
= b, e = , ().
|
|
. , a/e , . : .
. r - , d , r / d , .
. .
y
a/e d
M(x, y)
r1
0 a F1 x
OF1 = c
:
a/e + d = x, d = x a/e.
(x c)2 + y2 = r2
: , b2 = c2 a2:
.. /a = e, r = ex a.
: .
. .
. , .
: c2 = a2 b2.
: c2 = a2 + b2.
: .
. , 2,
c2 = 25 9 = 16.
: c2 = a2 + b2 = 16, e = c/a = 2; c = 2a; c2 = 4a2; a2 = 4;
b2 = 16 4 = 12.
: - .
.
. , , , , .
.
(, )
Π F x
p/2 p/2
( ) . .
: AM = MF; AM = x + p/2;
MF2 = y2 + (x p/2)2
(x + p/2)2 = y2 + (x p/2)2
x2 +xp + p2/4 = y2 + x2 xp + p2/4
y2 = 2px
: x = -p/2.
. 2 = 8 , 4.
, = 4.
r = x + p/2 = 4; :
x = 2; y2 = 16; y = 4. : M1(2; 4), M2(2; -4).
.
, .
. , . , .
|
|
.
.
. , l .
- , , . . j .
r
r =
j
0
l
, , .
:
x = rcos j; y = rsin j; x2 + y2 = r2
. :
. , , . .
: ;
. , 1/2 , a 3/2, b , = = 1/2. = /a = 1/3. F1(0; 0) F2(1; 0).
y
F1 F2
-1 0 ½ 1 2 x
-
. :
. , , . .
, .
. , 5 , 4, b 3, c2 = a2 + b2; c = 5; e = c/a = 5/4.
F1(-10; 0), F2(0; 0).
.
y
3
F1 -9 -5 -1 0 F2 x
|
|
-3
.
.
, , , :
F(x, y, z) = 0.
.
, . , - .
F(x, y, z) = 0 (x, y, z) = 0 , L.
.