skas
iarxeio
pro.blogspot.gr
1
= +
-
- -
+ + + ... +=
+ + + ... +
- + + =
= = -1
+ ++ = -
+ = -
:
i 5 = i 41+1 = i , i 9 = i42+1 = i ,.., i 4-3 = i 4( -1)+1 = i
z 2 = (x + xi)2 = x2 + 2x2i - x2 = 2x2i
:
z 4 + z 8 + z12 + ... + z 4
i z 2 + i 5z 6 + i 9z10 + ... + i 4-3z 4-2 =
z2 (z2 + z6 + z10 + ... + z4-2 )=
i (z2 + z 6 + z 10 + ... + z4-2 ) =
z 2 2x2i 2= = = 2x2 = Im(z )i i
2
:z3 = z z2 = (x + xi) 2x2i = -2x3 + 2x3i
:
z 3 - z2 + z + i = 0 (-2x 3 + 2x 3i) - 2x 2i + (x - xi) + i = 0
- 2x3 + x = 0( x3 x ) ( x3 x2 x 1) i 0
x3 x2 x 1 0 -2 + + 2 - 2 - + =
2 - 2 - + =
2x3-x = 0 2x3-x = 0 x (2x2 -1) = 0
( x3 x) x2 1 0 x2 1 0 x2 1 0
2
-
- 2
+
=
- 2
+
=
2 - =
2x2 -1= 0 2x2 =1 x2 = 1
x = 2
2 2:
z1 = 2
(1+ i)2
z2 = - 2
(1+ i)2
:
4 2z1 (z1) (1 i) (1 i 1) 4i 12
= 2 = 1 + 2 =
1 + 2 - 2 = 1 2 = - 2 4 4
(1)
z2 = - 2
(1+ i) = -z12(2)
:
4 4 42 1 1
(1)
z = (-z) = z = -1
:2 2 3 2 3 2
1 1 2 2 1 1 2 2
2 1 1 2
(2)z + z+
z + z=
z + z + z + z=
z z zz
3 2 3 2
= z1 +z1 + (-z1) + (-z1) =
z1(- z1)3 2 3 2 2z1 z1 z1 z1 2z1
2 21 1
2z z
+ - += = = -
- -
:
z12014 z1
4503+2 (z14 )503
z12
(1)
( 1)503 z12 z1
2= = = - = - =
2
(1 i) (1 i 1) i i 2 1 1
= - + = - + 2 - = - 2 = - 2 2 2 (3)
2014 (2)
2014 2014 (3)
z2 = (-z1) =z1 = - i
:2014 2014z1 + z2 = (-i)+ (-i) = -2i
3
- + =
= + =
=
4w 2 -2w +1 = 0 D = (-2)2 - 4 4 1 = -12 < 0
: w = 1
i3
4 4
z1
2
= wz
. z1 = 1 z1 0 , w 0 , (2)
:
4w + 1
=2 4w 2 -2w + 1 =0 w =
1
i
3
z1 = 1
i
3
w 4 4 z 2 4 4
z1 =
1 i
3=
1,
z2 4 4 2
(1 1)z1
2
z 2 2z 2= =
, 0 , z1 z 2 , :
2 2 2(O) = z1 = 1 = 1, (OB)2 = z2
2 = 22 = 4
:2 2 2 2 2 24z1 + z2 = 2z1z2 (z1- z2 ) = -3z1 |z1- z2| = 3 |z1| |z1- z2|= 3
:2() = 3 () = 3
:
(OB)2 =(O)2 +(B)2 ,
.
O=900
(OA) = (OB)
2,
OBA 300
= AOB 600
=
4
+ + - = +
- =
- = + -
:
2 2 ( )( )w1 w2 w1 w2 w1 w2 w1 w2 ( w1 w2 ) (w1 w2 )+ + - = + + + - - =
= ( w1 + w2 )( w1 + w2 ) + ( w1 - w2 )( w1 - w2 ) =
= w1w1 + w1w2 + w1w2 + w2w2 + w1w1 - w1w2 - w1w 2 + w 2w 2 =
2 2= 2w1w1 + 2w2w2 = 2w1 + 2w2
z = x + yi, x ,yR M(x, y)
z -2i =1 z-( 0 +2i ) =1 ()=1, (z) z (0 ,2)z (0,2)
1
z
(0,2) =1, z -2i =1
z1, z 2 z -2i =1,:
z1 - 2i =1 z2 - 2i =1
w1 = z1 - 2i w 2 =z2 - 2i ()
:
(z1 - 2i) + ( z2 - 2i)2 + (z1 - 2i) - ( z2 - 2i)
2 = 2 z1 - 2i
2 + 2 z2 - 2i2
2 2 2 2z1 + z2 - 4i + z1 - z2 =2 1 + 2 1
2z1 + z2 - 4i +1= 4
z1 + z2 - 4i = 3
5
( - ) = ( - )
= +
=
= -
-
(1):
(3z -2i)2013 = (iz -6)2013 (3z-2i) 2013= ( iz -6) 2013
( 3z - 2i) = (i z - 6) ( 3z - 2i) 2 = (i z - 6) 2
( )( ) ( )( ) 3 z - 2 i 3 z + 2 i = iz -6 - iz -6
9 z 2 + 6iz - 6i z + 4 = z 2 + 6iz - 6i z + 36
8 z 2 = 32 z 2 = 4 z = 2
:
z = 2 z 2 = 4 z z = 4 4
= z (3)z
z = x + yi x ,yR ,:
42014 (3)
u = z - = (z- z)2014 =(2 yi)2014 = z
= (2 y)2014 i2014 = (2 y)2014 i2 = - (2 y)2014 R
u R
:
2014 2014 ( )2014 4028(3)2014
u = z - 4 = (z- z) = z- z z + z = 2 z
:
u 24028
6
w1 = 1+ 2 z1 i w 2 = 1+ 2 z2 i, z1, z 2
(1), z1 = z2 = 2 . z1 , z2 O(0, 0) r = 2 , z1 -z2 4
:
w1 - w 2 = 1+ 2z1i -1- 2z2i = 2i (z1 - z2 ) =
= 2 i z1 - z2 = 2 z1 - z2 = 2 z1 - z2 2 4 = 8
- + = + + + =
+ + +
- + +
+ +
:
|z 2 - 2zi| +2|z| = 2+ |z + 2i|
|z| |z - 2i| + 2|z| = 2+ |z - 2i|
| z | | z - 2i | - | z - 2i | +2 | z | -2 = 0
( ) (|z - 2i| |z| -1 + 2 |z| -1) = 0
(| z | -1)(| z - 2i | +2 ) = 0 |z| = 1 (1), | z - 2i | +2 0 z C: x2 + y2 =1 (0,0) r =1
w :
|w +3z | + | 2w +5z |= |w +3z | +2w + 5 z2
|w +3z |+ w + 5 z (w +3z) - (w + 5 z) = 1 | z |= 1 1= 12 2 2 2 2
1|w + 3z| + |2w + 5z|
2
7
:(1)
22| z -1 + z +1| | z +1 + z | +1 = 4.:
| z -1 + z2 +1| | z -1+ z2 +1| = | z(z +1) | = | z | | z +1 | (2)
| z |=1 (2) | z -1 + z2+1| | z +1| (3)
| z -1 + z2+1| | z -1| (4)
z = x + yi x,y[ -1,1] : | z | =1| z |2 =1 x 2 + y 2 =1 (5)
(5)
2 2 2 2| z +1| = (x +1) + y = x + y + 2x +1 = 1+ 2x +1= 2(1+x) = 2 1+x (6)
(5)
2 2 2 2| z -1| = (x -1) + y = x + y - 2x +1 = 1- 2x +1= 2(1-x) = 2 1-x (7)
(3) (6) :
| z -1 + z2+1| 2 1+x (8)
(4) (7) :
| z -1 + z2+1| 2 1-x (9)
x[ 0 ,1] (8):| z -1 + z2+1| 2
x[ -1, 0] (9):| z -1 + z2+1| 2
:2 |z - 1 | + |z 2 + 1 | 4
zC | z | =1 :
|z 2 + uz + v|< 1 (10)
(10) z =1 z = -1:|1+ u + v | | 2+2v|2>2|1+ v|| v+1|2|-1+ v|| v-1|| v+1|+ | v-1| | (v+1)- (v-1)| 2 >2
.
zo C |zo | =1,2|zo + uzo + v|1
8
-
- - =
- + = + -
- - + +
= ( - ) - - - - -
:
112
3
2
13i
22
1
2
3i1
2
3i1 25
2522252525
==
-+
=-=
-=
-
( )25
z 3 3i 2 1 i 3
z 3 3i 22
- - - = - + =
z ( 3 , 3) = 2 z z) x Re( - k , x = 3
( , ) = 2:
Re(z)- x x - Re(z) x +
3- 2 Re(z) 3+ 2 1 Re(z) 5
Re(z) 1 z =1 + 3i , 5 z = 5 + 3i
:
z (3 + 3i) = 2z=x+yi
- (x - 3)2 + (y - 3)2 = 4
(x - 3)2 4 x -3 2 -2 x - 3 2 1 x 5
Re(z) 1 z =1+3i , 5 z =5 +3i
9
:
2 w -3+3i = iz +3-3i 2 (w -3-3i) = i(z -3-3i) )
2w -3-3i = i z -3-3i 2w -3-3i =2 w -3-3i =1 w - (3+3i) =1 w ( 3 , 3) r1 =1
z w () ():
z - w = ( z - (3+3i)) - ( w - (3+3i)) = u - v
z (3 3i)u = - + v = w - (3+ 3i) u =2 v =1:
2 1 3vuvuwz - = - + = + =
2 1 1vuvuwz - = - - = - =
3w1 z -
.
:z + w = (z - (3+ 3i)) + ( w - (3+ 3i)) + 6(1 + i) = u + v +6(1+ i)
:
3262612i)6(1vui)6(1vuwz + = + + + + + + = + + = +
( ) ( )z + w = u + v +6(1 + i) 6(1 + i) -u + v 6(1 + i) - u +v = 6 2 -3 = 6 2 -3
6 2 3wz36 2 - + +
10
:
) ( ) 6 2 3(MNw)(zwz + = - - = B = +
) 6 2 3()(MNw)(zwz + = - - = AG = - ,
u = z - (3+ 3i), :23i)(3u = z - + = (1)
u = 2 u 2 = 4 u u = 4 u = 4 (2)
u z1 , z2 z,:
( 1 2 )1 2
1 1 k = z - z - =
z - 3 - 3i z - 3 - 3i
(( 1 ) ( 2 ))1 2
1 1 = z -3-3i - z -3-3i - = z -3-3i z -3-3i
( ) ( )( 2 )
1 2 1 21 2 1 2
1 1 1 4 4 = u - u - = u - u - = u u 4 u u
= 1 ( u1 -u2 )( u1 - u2 ) =
1 ( u1 - u2 )( u1 - u2 ) = 1 u1 - u2 24 4 4 k :
k = 1u1 - u2
2 0 4
k1
u1 u22 1 ( u1 u2 )
2 (1)1(2 2)2 4
4 4 4= - + = + =
0 k 4
+ = +
=
+ > <
- + + + =
11
:
iz +8 = 2 2iz +1 iz +8 2 = 4 2iz +1 2
(iz + 8)(iz + 8) = 4 (2 iz + 1)(2 iz + 1)
(iz + 8)(-iz + 8) = 4(2iz +1)(-2iz +1)
z2 + 8iz- 8iz + 64 =16 z 2 + 8iz- 8iz + 4
15z2 = 60 z 2 = 4 z = 2
z (0,0) = 2
:z1 = 2 z2 = 2
:
1 2 1 2 1 2 1 2z + x z > 3 z + x z2 > 3 (z + x z ) (z + x z ) > 3
1 2 1 2 1 1 2 1 2 2(z + x z ) ( z + xz ) > 3 z2 + x z z + x z z + x 2 z
2 > 3
1 2 1 2 1 2 1 24x2 + (z z + z z ) x +1 > 0 4x2 + (z z + z z ) x +1 > 0
1 24x2 + 2Re(zz ) x +1 > 0 , xR
4x2 + 2Re(z1z2 ) x +1 :
0 ( 2 -1)( z 2 -4) > 0 (3)
( -1,1) 2 -1< 0 (3) :
z 2 - 4 < 0 z 2< 4 z < 2 ,, z > 2
12
z 2 ,.
z = 2 , z,
(1)C : x 2 + y 2 = 4 , () .
z = 0 , (1) 16 = 0 ,. z 0 :
z2 0
z4 2z3 8z 16 0
z2 28 16 2 16 4zz z2
0 zz2
zz
0 (4)+ + + = + + + = + + 2a + =
z + 4
= u (5), :z
4 2 2 4 16 2 2 16 2z u z z 2 u z 2 u 8 ,z z z z +
2
= + 2 + = + = -
(4) :
(u2- 8) + 2au =0 u2+ 2au -8 =0 (6):
D = 42 +32 = 4(2 +8) >0 (6) :
2 8 2u = -
2a
2
+
= - a +82
(7)
(5) :
( )(7)z2 - uz + 4 = 0 z2 + a 2 +8 z + 4 = 0 (8):
( ) ( )2D = a 2 +8 -16 = 2 +2 +8 2a 2 +8 -16 = 2 2 -4 a 2 +8 -1
13
y = 0 , z = x + 0 i = xR () z =2z= 2
z = 2 :
24 + 2 23 + 8 2 + 16 = 0 = -1,, (-1,1)
z = - 2 :
(-2)4 + 2 (-2)3 + 8 (-2) + 16 = 0 =1,, (-1,1)
y 0
x = 0 , z = 0+ yi = yi I () z =2z= 2i
z = 2i :
(2i)4 + 2 (2i)3 + 8 2i + 16 = 0
24 i4 + 2 23i3 +8 2i + 16 = 0
16-16i + 16i + 16 = 0 32 = 0 ,.
z = - 2i :
(-2i)4 + 2 (-2i)3 + 8 (-2i) + 16 = 0
24 i4 + 2 (-2)3 i3 -8 2i + 16 = 0
16 +16i -16i +16 = 0 32 = 0 ,. x 0 (1) x + yi xy 0
[ ]= =
- =
=
= -
- [ ]
z = x + yi, x , yR M(x , y).
:
z -2i = z x +yi -2i = x +yi x +(y-2)i = x +yi
x2 + (y - 2)2 = x2 + y 2 x2 + (y - 2)2 = x2 + y 2
x2 + y2 -4y+ 4 = x2 + y2 -4y+ 4 = 0 4y = 4 y =1
14
z y =1 , z = x + i , x R:
w z2 1
(x i)2 1
(x i)2 x -i
z x i (x i)(x i)= - = + - = + - =
+ + -
2 x - i 2 x 1= x + 2xi -1- 2 =
x -1-
2
+
2x +
2
i
x +1 x +1 x +1 :
32w R 2x + 1
= 0 2x + 2x +1 = 0 (1)x +1
(1) x R, x=Re(z)
g(x ) =2x 3 + 2x +1 , xR
g g(x)=6x2 + 2,xR
x R g(x) > 0 , g R
:
g R
( )x x xlimg(x) = lim 2x3 + 2x +1 = lim 2x3 = - - - -
( )x x xlimg(x) = lim 2x3 + 2x +1 = lim 2x3 = + + + +
g g(R) = R, R,
g( ) = 0 , g.
z 2 = x2 +1 , -1 f (x2+1) 1
f [1,2 f(1) < f(2) , f] [1 , 2] [ ]Re(z) 0,1 ,:
2 2f
0 x 1 0 x 1 1 x +1 2
f (1) f (x2 +1) f (2) -1 f (x2 +1) 1
- 2 = 2 + 2 -
= = + = - +
15
2 + 2 = =
= - -
- = - =
= +
:2 ( ) ( )z-w = z-w z-w = ( z-w)( z -w) = zz - zw - zw + ww == z 2+ w 2-(zw+ zw) = z 2+ w 2-( zw+ zw) = z 2+ w 2-2Re(zw) (1)
:
z2 w
2 2Re zw) z
2 w
2 2Re zw) 0
(1)
+ = ( + - ( =
z - w 2 = 0 z - w = 0 z - w = 0 z = w
:
1
1
f (2) = 3 f (2) = 3z =w 3+ f (2)i =f -(2)+3i
f(3) 2f -(2) 3
== f ,
. f ,:
f
2 < 3f (2) < f (3) 3 < 2. f .
1 2x, x R f
x1 f (x2)- f (x1)< - f (x2):
f (+)x1< x2 2x1< 2x2 2x1 - f (x1) < 2x2 - f (x2) - f (x1) < - f (x2 ) - f (x1) < - f (x2 )
f ( 2x1 - f (x1)) > f ( 2x2 - f (x2 )):
f ( 2x1 - f(x1)) > f ( 2x2 - f(x2)) (+) f (2x1 f (x1)) x1 f (2x2 f (x2)) x2 g(x1)>g(x2)x1 x2
- - > - - - > -
g .
xR :
( ) ( )f -1(x) =2x-f(x) f 2x-f(x) =x f 2x-f(x) - x = 0 g(x) = 0
f R,
[1,4 , 1 (1 ,4) ,]
16
1 1 1
f (4) f (1) f (4) f (1)f ( ) f 3f f (4) f (1)
4 1 3=
- ( ) =
- ( ) = -
-(2)
xR :
g(x) = f ( 2x- f (x)) - x
2x-f(x) R,,
f (2x- f(x)) R, g R,
.
g
[ 2 ,3] , 2 (2,3) ,
2 2g g(3) g(2)
g g(3) g(2)3 2
( ) = -
( ) = --
(3)
x=2 g(2) = f ( 2 2- f (2)) - 2 = f (4-3)- 2 = f (1)- 2 , f (1) -g(2) =2 (4)
x=3 g(3) = f ( 2 3- f (3)) -3 = f (6- 2) -3 = f (4) -3 , f (4) - g(3) =3 (5)
(2) (3) :
( ) ( )3f (1) - g( 2) = f (4)-f (1) - g(3)-g(2)
( ) ( )(4),(5)
3f (1) - g ( 2) = f (4)-g(3) - f (1)-g(2)
3f (1) - g ( 2) = 3 - 2 3f (1) = g ( 2) +1
- + =
+ =
p q -
pq = > *
[ - ]>
17
:
D = ( -6 sunq)2 - 4( 5sun2q + 4) = 36 sun2q - 20 sun2q -16 = -16 (1 - sun2q) = -16hm2q:
1,2z 6sunq i4hmq
3 2 i2
= = sunq hmq
z1,2 = x +yi ,x,yR , :
x =x 3 3
yy 22
sunq = sunq
= hmq = hmq
(1)
:2 2(1)
2 2 1x y
19 4
sun q + hm q = + =
z1,z2 x2 y2
+ =19 4
,
qR
:
z1 - z2 = 4ihmq = 4hmq , 0 4p
q
:
0p
0 2
0 4 2 24 2
q hmq hmq
z1 - z2 2 2, z1 - z2 2 2
2
pq = z1 =2i, z1
n = 2 n in
n = 4k , k N * z1n = 2 n R
n = 4k +1, k N* z1n = 2 n iR
n =4k + 2 , kN* z1n = -2 n R
n =4k +3 , kN* z1n = -2 n iR
n = 2m, m N*
:2 2 2 ( 2 ) y 0x y
1 4x2 9y2 36 y2 36- 4x y2
4 9- x > y2
9 x29 4 9 9 3
+ = + = = = = -
18
M ( x, y) C ,:
( ) ( ) ( ) ( ) 22
d M,A = x - 2 2 + y -3 2 = x - 2 2 + 2
9 - x -3 3
( ) 22
f(x)= x-2 2 +2
9-x -33 [ ]-1,2
,
x1, x2 [ -1,2 ,:]m = f(x1) M = f(x2 ) , m f ( x ) M
C ,.
- = ( ) + - - - =
=
=
- + =
- - =
= ( ) -1
2 2 2x +yi -1 =x +1 (x -1) +y =x +1 (x -1) +y = x +1
x2 - 2x +1+ y2 = x2 + 2x +1y2 = 4x, x 0
z y2 = 4x
w = +i ,R (2) :2 2 2 2 ( 2 2 ) + i - 2 - + i - 2i = 8 ( - 2) + - + ( - 2) = 8
2 - 4 + 4+2 -2 -2 + 4 - 4 = 8 -4 + 4 -8 = 0 - + 2 = 0
w x-y+2=0
19
u = x + yi x, yR (3) :2 u0 2 2(x yi) 2x 2y
2 2 2 2 2 2w = +i = +i = -
= = -
u x + yi x + y x + y x + y
- + 2 = 0 ,:
2 22 2 2 2
2x 2y+ +2 =0 2x +2y +2x+2y =0
x + y x + y2 2
x 2 + y2 +
x +
y =
0
x + 1 +
y + 1 =
1 2 2 2
x = y = 0 , u = 0 ,.
u K - 1,-
1 2 2 2
=2
,
O(0,0)
( x1, y1 ) ,:(yy1 =2 x + x1 ) 2x - y1y+2x1 =0
1
= 2
y
():
11
=1 2
=1 y = 2y
x1 =1
( )M1,2 ,(),().
( x, y) ,:
( )
y22 2
y 2x-y+2 4
- + y -4y+8 y -4y+8d M, = = = =
1+1 2 4 2 4 2
2f(y) = y -4y+8, yR. f R yR:
f (y) = 2y - 4:
f (y) =0 2y - 4 =0 y =2 f (y) >0 2y - 4 >0 y >2
f (y) f .
x - 2 +
f (y) - 0 +f (y)
.
f y =2 . y=2 x =1, ( )M 1,2().
20
:
y 0
x > 0 (5) :
x2 x+ 5 - 2 = 0 x x + 5 x - 2 = 0
x x
g(x) =x x +5 x -2, x >0
g ( 0,+ ) g (x) = 3 x + 5 , x >02 2 x
:
g (x) >0 x ( 0,+)
g ( ) = 0,+:
x 0 ( )
x 0 ( )limg x = lim x x +5 x - 2 = -2
+ +
x ( )
x ( )limg x = lim x x +5 x - 2 = +
+ +
g ( )g() = -2,+ 0g() g , g(x) = 0
( )A = 0,+ z z = w Im ( z)
, + =
= --
21
0 , = 0, 2
22
e -x > 0 x > 0 , (x) > 0 (0, + ) .
(0) = -1
lim (x) lim x3 x2 x e -x x x
= + + - = ++ +
limx3
x2
x = lim x3 = +
x x + +
+ +lim e -x =0
x +
([0, + )) = [-1, + ). 0 (0) 0 , xo (0 , ) +
, , j(xo ) = 0
xo (0 , ) + , f (xo ) g(xo ) = a (1)
h h = (0 , xo)v (xo , +) . h
(1)
h(x)=g(x)f (xo ) - = g(x)f (xo ) -g(xo )f (xo ) =g (x) -g(xo )f (xo )(x - xo ) f (xo )(x - xo ) x - xo
g(x)(x xo ) g(x) g(xo ) 1 g(x) g(xo ) h(x) = -
-
2
+ =
g(x) -
-(x - xo ) x - xo
x - xo
(2)
x > xo, [ xo ,x] g
... 1 (xo ,x) , g(1) = g(x) - g(xo )
x - xo (3)
(2) (3) :
h(x) = x
1x
( g(x) -g(1))- o g
g(x) = ex +xex =(x+1)ex
g(x) = ex + (x +1)ex =(x+2)ex >0 x > xo > 0
g , x > x1 g(x) > g(1)
o1h(x) = x
1x
(g(x) -g( )) >0-
x > xo > 0
h (xo , + )
0 < x < xo , [x ,x ]o g
... 2 (x, xo) , g(2) = g(x)-g(xo )
x - xo (4)
23
(2) (4) :
h(x) = x
1x
( g(x) -g(2))- o g
g(x) = ex +xex =(x+1)ex
g(x) = ex + (x +1)ex =(x+2)ex >0 x (0 , xo)
g , x< x2 g(x)0-
x (0 , xo)
h (0 ,xo )
+ =
=
+ +
+ + = = =
+
:
zw + 2i = z w
zw + 2i 2 = z 2 w 2
(zw + 2i) ( zw + 2i) = z 2 w 2 (zw + 2i) ( zw + 2i) = z 2 w 2 (zw +2i) ( zw - 2i) = z 2 w 2
z2 w 2 + 2i zw - 2i zw + 4 = z 2 w 2
2i (zw - zw) = -4 2i ( zw - zw ) = -4 2i 2i Im(zw) = -4 -4 Im(zw) = -4 Im(zw) =1
24
:
zw + 2i = z w zw + 2i = z w
zw=uzw - 2i = zw u - 2i = u u - (0 +2i) = u - (0 +0i)
u A
(0, 0) A(0,2), y =1, Im(u) =1Im(zw) =1
:
Im(zw) =1
:
zw = x + i xR (1)
(1) z 0 w = x + i
xR,:z
2x + i x +1x + iw = w = w =
z z z(2)
:(2 )
z2 + w 2 + Re( zw) 3
x 2 12z 2 x 3
z+
+ +
z4 + x2 +1+ x z 2 3 z 2
( )2y z
z4 x 3 z
2 x2 1 0
=
+ - + +
y2 + ( x - 3 ) y + x2 +1 0 (3)
y2 + ( x - 3) y + x2 +1 y :
( )2 = x - 3 - 4(x2 +1) = x2 - 2 3x + 3- 4x2 - 4 == -3x2 - 2 3x-1= -(3x2 +2 3x+1) = -( 3x+1)2 0
(3) x,yR, z 2 + w 2 + Re(zw) 3
< 0,() z 2 + w 2 + Re(zw) > 3,,= 0
25
:
0 ( 3x 1)2 0 3x 1 0 x 1
x 3
33= - + = + = = - = - (4)
:
( )z 2 + w 2 + Re(zw) = 3 y2 + x - 3 y + x2 +1= 0= 0 ,:
2(4) y z 2
3
y-x + 3
y 3+ 3
y 2 3 =
z 2 3
2 2 3 3= = = = (5)
(2) :
(2),(5)21
w2 x +1
w2 3
+1w
2 2w
2 2 3
32 3 3z2
3
= = = = (6)
(5) (6) : z2 = w 2 =
2 3
3
:
z2w
2 z
2w
2
z2w
2 Re(zw) z
2 w
2 Re(zw)
= =+ +
zw2
(1) x2 +1 x2 +1= zw
2 + Re(zw)= x2 +1+ x
= x2 + x +1
x2 1
f (x) = 2 +
,x + x +1
xR
xR:
2x(x2 x 1) (x2 1)(2x 1) x2 12 2 2 2f (x) (x x 1) (x x 1)
+ + - + + -= =
+ + + +
:
f (x) = 0 x2 -1= 0 x2 =1 x= 1
f (x) >0 x2-1>0 x2 >1 x >1 x < -1 x >1
:
( 1)2 1 2
f ( 1) 2 2( 1) ( 1) 1 1
- +- = = =
- + - +
12 1 2
2f (1) 1 1 1 3
+= =
+ +
x x x
x2 1 x2limf (x) = lim 2
+ = lim 2 =1 x + x +1 x
26
f (x) f
.
f x1 = -1 f(-1) =2 x2 =1
f (1) = 2
, x R 2
3 f(x) 2
3
z
2w
2
z2w
2 +Re(zw)
2
3 2
+ - + - + =
+ =
= + -
+ + = + +
- - + - + - +
+
- =
- =
z = x + yi x ,yR :
z + 2 - 4i 2 + z - 2 + 4i 2 = 58
x - -1 1 +
f (x) + 0 - 0 +
f(x)
.
2 2
3
.
1
1
27
x + yi + 2 - 4i 2 + x + yi - 2 + 4i 2 = 58
x + 2 + (y - 4)i 2 + x - 2 + (y + 4)i 2 = 58
(x + 2)2 + y - 4)2 + (x - 2)2 + (y + 4)2 = 58
x2 + 4x + 4+ y2 -8y +16+ x2 - 4x + 4+ y2 +8y +16 = 58
2x2 + 2y2 =18 x2 + y2 = 9 C: x2 + y2 = 32
z O(0,0) = 3 x =0 :
02 + y2 = 32 y2 = 32 y = 3 y = -3 yy A(0,3) B(0, 3)-
z O(0,0) = 3 ,:
z = 3 z 2 = 9 z z = 9 z = 9
z = 9
z z(1)
:9 (1) 9
w =z + = + z =wz z
w .:
9 (1)w = z + = z + z = 2Re(z) (2)
z
z = 3 ,(2)
-3 Re(z) 3 -6 2Re(z) 6 - 6 w 6
z1 ,z2 z3 :
1 1 11
9zz = 9 z = ,
z 2 2 2 2
9z z = 9 z =
z 3 3 3 3
9z z = 9 z =
z
:
1 2 3 1 2 3 1 2 31 2 3
9 9 9z +z +z = z +z +z = z +z +z = + + =
z z z
z2z3 z1z3 z1z2 z2z3 z1z3 z1z2
1 2 3 1 2 3 1 2 3
= 9
1+
1+
1=
9
+ + =
9
+ + =
z z z zz z z z z
= 9 z2z3 +z1z3 +z1z2 =
z2z3 +z1z3 +z1z23 3 3 3
:
z2z3 + z1z3 + z1z2z1 + z2 + z3 = 3:
z2z3 +z1z3 +z1z2 = 3 z1 + z2 + z3
28
:
vsunq -ihmq = z(w - z) + z(w - z) + 2z 2 - w 2 + 3
vsunq - ihmq = z w - z z + z w - z z + 2z z - w 2 + 3
vsunq- ihmq = z w + z w - w 2 +3
vsunq- ihmq = z w + z w - w w +3
vsunq - i hmq =w (z + z -w) +3
vsunq - i hmq = 2x ( 2x -2x) + 3
vsunq - ihmq = 3 vsunq = 3 + ihmq (3)
R +
, Z sunq 2
0 ,
(3) sunq :
v = 3
+ ihmq
v =
3+ iejq
sunq sunq sunq
v = a + bi , R, :
3 1
3
a= a =
sunq sunq ejq = b
:
2 , R 21
=1+ ej qsun q
+ , Z
2
:
2 2a 1 2 a 2 1
9 9= + b -b =
v x2
C : - y2 =19
M(a,b) x2
C : - y2 =19
A(0,3) M(a,b) :
d(A,M) = d = (a - 0)2 + (b -3)2 = a2 + (b -3)2 = a2 + b2 - 6b +9 (4)
M(a,b)C :2 2a 2 1
a 1 2 2 9 9 29 9
-b = = + b a = + b
a2 (4) :
d = 9 + 9b2 + b2 - 6b + 9 = 10b2 - 6b +18
d(b) = 10b2 - 6b +18 , R
29
R :
(10 2 6 18) 20 6 10 32 2 2
d( )2 10 6 18 2 10 6 18 10 6 18
b = b - b + = b - = b -
b - b + b - b + b - b +
:
d (b) =0 10b-3=0 10b = 3 b = 310
d (b) >0 10b-3>0 10b > 3 b > 310
d (b) d .
d = 3
10
= 3
10
:
2 2 2a -
3 =1 a
=1+ 9
a2 =
9
109
a
=
3 1099 10 9 100 100 10
A(0,3) :
1
3 109 3 ,
10 10M
M2
-
3 109, 3
10 10
= + -
( + ) - = =
( - ) = =
=
- +
- 3
10+
d (b) - 0 +d(b)
30
>
=
( )
:
(1+ 2i)w -5 = 3 5 (1+ 2i)w - 5 = 3 5 1+ 2i
1+ 2i w - 5(1-
2i)
= 3 5 w - (1- 2i) = 35
w
(1, - 2) R = 3
:z=x+yi
f(z) =f(z) z +1+i = z +1-i z -z = 2i 2yi = 2i y = 1z y =1
y =1,-2 -1
d(K,e) = = 3 = R.1
w z z = w
w = x + yi
(1, - 2) R =3 , (x -1)2 +(y +2)2 = 9 y = 0 :
(x -1)2 = 5 x = 5 +1 x = - 5 +1 w ,:
w = 5 +1+ 0i = 5 +1 w = - 5 +1+ 0i = - 5 +1
w- (5 - 6i) w (A 5,-6) , 5 - 6i w- (5 - 6i) :
2 2w 5 6imax
(K ) R (5 1) + ( 6 2) 3- + = A + = - - + + =
= 16 +16 + 3 = 32 + 3 = 4 2 + 3
q z xx ,:
y y=1 1
x xefq = efq = , x >0
31
x t :1
efq(t) =x(t)
t :
2 2
1
q (t) =
-
x (t)
sun q(t) x (t)(1)
:
22
1= 1+ e j q(t)
sun q(t)(2)
(1) (2) :
( 1 2 (t)) (t) x2(t)x (t)
+ ej q q = - (3)
to M (3,1),
t = to (3) :
( 1 2 (to )) (to ) x2(to )x (to )
+ ej q q = - (4)
:x(to ) = 3 , x (to ) = 1
2
o 2o
1 101+ (t ) 1 1 1ef q = +
x (t )=
+
9=
9 (4) :
o o
10 q(t ) = -
1 q(t ) = -
1rad/s
9 9 10 M yy z xx .
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