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C(a, b)
R
M(x, y)
R = |CM| =
(x a)2 + (y b)2
(x a)2 + (y b)2 = R2
R
O(0, 0)
x
2
+ y2
= R2
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F1, F2
|F1F2| F1, F2 Ox
M(x, y)
O
yy y
GGx
mmmmmmmmmmmmmmmmmmmmmmmm ***
****
****
F1(
c, 0) F2(c, 0)
r1 r2
M(x,y)
|F1M|= r1, |F2M|= r2 r1 =
(x + c)2 + y2, r2 =
(x c)2 + y2.
r1 + r2 = 2a
a
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(x + c)2 + y2 +(x c)2 + y2 = 2a
a2
c2 = b2 (2a > 2c)
x2
a2+
y2
b2= 1
(a; 0), (a; 0), (0; b), (0; b), a x a; b y b.
x, y
Ox, Oy
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yyy
D(0, b)
C(0, b)
GGx
A(a,0) B(a, 0)
mmmmmmmmmmmmmmmmmmmmmmmm ***
****
****
F1(c, 0) F2(c, 0)
r1 r2
_________M(x,y)
d
x = ax = a
AB
CD
r1 = a +c
a x, r2 = a c
a x c
a=
r1 = a + x , r2 = a x
=2c
2a =c
a, 0 < 1.
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a
x = a
r
d
r
d
=
r
d
0 2a)
b2x2 a2y2 = a2b2
x2
a2 y2
b2= 1
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x, y
Ox,Oy
y = 0
x = a
Ox
A(a; 0), B(a; 0)
x = 0
y2
b2 = 1
Oy
a
b
y =b
ax2 a2
x = a
y = 0
x
y
Y =
b
a x
Y y = ba
x ba
x2 a2 = ab
x +
x2 a2
x
Y
y
0.
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Y = ba
x
O
yy y
GGx
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
OO
OOOOOO
OOOOOO
OOOOOO
OOOOOOOOO
OOOOOO
OOOOOO
OOOOOO
OOOOOO
OOOOOO
OOOOO
ffffffffffffffffffffffffffffffffffffff
_________
_________
y = ab
xy = ab
xd2
r2r1b
bA
F2(c; 0)BF1(c; 0)x = a
x = a
=2c
2a=
c
a
.b2
a2
= 2
1
( > 1)
b
a
a = b
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r1 = x + ar2 = x a
r1 = (x + a)r2 = (x a)
a
x =a
x = a
r1 = d1, r2 = d2
16x29y2144 = 0
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8/14/2019 L5 101
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16x2 9y2 144 = 0 x2
9 y
2
16= 1 = a = 3, b = 4,
c =
a2 + b2 =
32 + 42 =
25 = 5 |a| = 3, |b| = 4, B(3;0), A(3; 0), = c
a=
5
3
y = ba
x = 43
x, d = a
= 35
3
= 95
|a| = 3, |b| = 4, B(3;0), A(3; 0), O(0;0),F1(5;0), F2(5;0), =
5
3, y = 4
3x, d = 9
5, y = 0
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p
O
yy y
GGx
K(
p2
; y)
x =
p2
M(x, y)
F(p2
; 0)
Fp
2; 0
, x = p2
M(x, y)
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|KM
|=
|M F
|
K
p
2; y
x+p
22
=
xp2
2
+ y2 y2 = 2px.
Ox
y2 0, p 0 x 0.
y =
2px
x
y
yy
y2 = 2pxy
GGx
-------
OF
M Kyyy
x2 = 2py
GGxmmmmmmm
OF M
K
yyy
x2 = 2py
GGx
OOOOOO
OOFM
K
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Ox
M(3, 4)
Ox M
y2 = 2px
M
42 = 2p3 = 6p = 16 p = 83
y2 = 2px = 28
3=
16
3x
y2 = 6x
y2 = 6x y2 = 2 3x
p = 3, F
32
; 0
, d = 32
,
p = 3, F
3
2; 0 , O(0;0), x =
3
2,
Ox, y = 0
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O
Ou
M
|OM| = , (M,Ou) =
,
M(, )
Ou
M(x, y)
O
yy y
GGx
ttttttttttttttttttttttttttt
_______
yM(x,y)
M(,)x u
GGmm
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x = cos
y = sin
=
x2 + y2
tg =y
x
A(2; 2)
=
4 + 4 = 2
2 , tg =22
= 1 1 = 4
, 2 =7
4
A(2; 2)
IV
= 74
A22; 74
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Oz
M
N M
N N(, )M z
M
Oz
z OM z
, , z
M(,,z) , 0 < , 0 < 2 , < z <
z
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M(x , y , z)
x = cos , y = sin , z = z
x = cos y = sin
z = z
= x2
+ y2
tg =y
xz = z
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M
N M
|OM
|= , (ON,Ou) = , (OM,Oz) = M(, , ).
0 < , 0 < 2 , 0 <
M(x, y, z)
x = sin cos
y = sin sin z = cos =
x2 + y2 + z2
cos =zx2 + y2 + z2tg = yx
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(O,i,j) (O,i,j )
O
yy y
GGx
wwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww
YYx777777777777777777
y
dd
___________
_______________
77
ww
ww
ww
w
b
y
a x
O
x
y
M
Oxy
M(x, y)
Oxy M(x, y)Oxy
O(a, b)
(Ox,
Ox) =
OM =
OO +
OM = ai + bj + xi + yj
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x = x cos y sin + a
y = x sin + y cos + b
x = (x a)cos + (y b)sin y = (x a)sin + (y b)cos
= 0
= 0 (i,i) = 0
x = x + ay = y + b
x = x
a
y = y b
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O = O
=
a = b = 0
x = x cos y sin y = x sin + y cos
x = x cos + y sin y = x sin + y cos
