1. Μιγαδικοί αριθμοί - Μαθηματικά κατεύθυνσης

of 70/70
ΜΑΘΗΜΑΤΙΚΑ ΘΕΤΙΚΗΣ - ΤΕΧΝΟΛΟΓΙΚΗΣ ΚΑΤΕΥΘΥΝΣΗΣ Μ Μ Ι Ι Γ Γ Α Α Δ Δ Ι Ι Κ Κ Ο Ο Ι Ι Α Α Ρ Ρ Ι Ι Θ Θ Μ Μ Ο Ο Ι Ι Γ / ΛΥΚΕΙΟΥ Δημήτρης Αργυράκης Γεράσιμος Θ. Κουτσανδρέας
  • date post

    29-Jul-2015
  • Category

    Documents

  • view

    328
  • download

    2

Embed Size (px)

Transcript of 1. Μιγαδικοί αριθμοί - Μαθηματικά κατεύθυνσης

/

-

.

.

- 1. z = + i, , R z = 0, = 0 = 0. 2. z = + i , 0,

1 1 1 = + i. z a b3. z = + i , R, Re (z) = . 4. z = x + (y - 1) i m (z) = 0, y = 1. 5. z1, z2 C Re (z1 + z2) = 0, Re (z1) + Re (z2) = 0. 6. yy. 7. i2 = - 1 i2003 = i. 8. xx. 9. z 0 z0 = 1. 10. 1, 2 z1 z2 xx 12,

z1 = z211. z1 = + i , z2 C, z1 + z2 = 2,

z2 = z1 .12. Re (z) = 2 z x = 2. 13. m (z + i) = 8 z y = 8. .

2

.

14. x2 - 2x + = 0, R, 1 + i 1 - i. 15. x2 + x + = 0, 0, , , R 2 + i

5 . 2+ i 16. x2 + x + = 0, , , , R* C. 17. Re (z1z2) = 0 Re (z1) .Re (z2) = 0.18. z - z = z . 19. z1, z2 C z1 + z2 = z1 + z2 . 20. z - z1 = z - z2 , z C,

(z1) B (z2).21. z - z1 = z - z2

z C z1, z2 C .22. z - z0 = , > 0

(z0) . 23. 2 + 3i z = 4.24. y = x z = + i R.y

25. .

2 2 0 2 2 4 6 x

3

.

z - 2 = 4.

1. x y : ) x - 2 + 2yi = - 2i + 2 - yi ) y + 2i = 3 - (2 + i) x ) 4y - 3yi - 2x = 2 - 5xi + 9i ) (x2 + 1) i + 2x = x2 - 2xi - 3 2. z = x2 - x - 9i w = 2 - y2i , x, y R. ) x, y z = w.

) x = -1 , z 2 .3. z = 6i - (3 - 4i) x - 3yi - (3i - 2) x + (4 - 2yi), x, y R. ) z + i. ) : i) Re (z) = 0 ii) z = 0 4. z = (2 + i) x + (y - 1) i - 5, x, y R. ) + i. ) z x, Im (z) = 0. ) x y, Re (z) = Im (z). 5.

z1 = 1 + i , z4 =

z2 =

1 1 + i, 2 3

z3 =

1 1 + i, 4 9

1 1 1 1 + i, z5 = + i , 8 27 16 54 100 z1 + z2 + z3 + z4 + z5 + z100 .

6. + i :

.

4

3 104 -47 7 12

2 2

.

) z1 = 2i + i

3i

+ i +3i

,

) z2 =

(1 - i )1+ i

+

(1 + i )1- i

.

7. + i :

) 3i (- 5i) ) 1 1- i )

) (2 + i) (- i + 3) 1- i - i+ 1 )

)

3 4i

(2 + 3i )(- i + 1)1 - 2i

8. + i :

) (2 - 3i) (4 - 5i) + 7i - 1 ) 3+ i 3 - 2i ) (1 - i)-3

1 - 2i 2i + 1 . ) i + 3 3i + 1

1 2 ) + i 2 2

2

9. ,

z1 = + i z2 =

12 + 8i 52 + 13i + . 2 - 3i 13i 12 + 5i . i

10. , : ( + i)2 =

11. x R : 1 + 2 2 i = 3

1 + xi . 1 - xi :

x, y R 12. z1 = x + 2y - i z2 = 11 - (4x - y) i . 13. z z -i

z3 - i = . z+ i14.

)

z z + (z - z ) = 3 + 2i .5

.

.

) z = z2.15. )

1 - i n+ 1 f () = . 1- i ) = i + i2+1 i , .16. (1 + i)20 = (1 - i)20. 17. ) .

) =

z R z . z+ i3 3

(1 - i ) (1 + i ) + ) u = , . n n (3 + i ) (3 - i )18. z = x + yi, x , y R.

z + 8i . z+ 6 ) x y, Im (w) = 0. ) x y, Re (w) = 0. ) () . ) . ) + i w =19. ) z2 + z + =0, , R 2 - i. i) . ii) .

) z1 , z2 z2 + z + 5 = 0 , z1 z2 + 5i = - 5i . z1 + z2 + i . 6

.

20. z = x + yi , x, y R, :

z2 + 2 z + 1 = 0.21. ) z = + ( - 1) i y = 4x + 1, R.

) z1 = 4+i z2 = (+3) - i , R z1.z2 R . z1 , z1+z2 R . ) z = ( + 1) + (2 3)i , R . z , R .22. 1 (2z). 2 (2 z ) , 3 (-2z) 4 (-2 z ). 1234.y 2 1 2 1 0 1 2 M(z) 1 2 3 4 x

23. z = 2 + i z1, z2 y = x - 2 y = 2x - 1. 24. :

2+ i ) z = 1 - 3i

) z =

(1 - i )1+ i

2

+ 2 - 4i

25. :

2+ i ) z = 1 - 3i

2

2 + i 5 ) z = 3 , . 7

n

.

.

26. z z + z = 2 + i. 27. z C z + 9 = 3 z + 1 , z = 3. 28. .

) w , w = - w . )

w=

z- 1 , z - 1, , z = 1. z+ 1

29. z z 3 z 4. ; 30.

1 = z- 1. z

z

:

z=

31. C :

z + z + 1 + i = 0.

1 32. z : 1 - z > z , Re (z) < . 233. z

2 z - 1 = z - 4

(0, 0) 2.

34. z

w=

z + 2i , z -1 . z+ 1

35. z ( ) :

- 2 Re (z) 2 .

(1)8

.

Im( z ) 2

(2) (3)

z 2

z .36. z :

) z + 1 - i = 3 ) z - 1 - i < 4 ) 1 < z - 1 + i < 237. z = x + yi, x, y R . ) , : i) (x - 3)2 + (y - 2)2 = 9 ii) 3x2 + 2y2 = 4

y 2 1 0 1 2 3 4 x K

iii) z - 3 + 2i = 4 iv) x2 + y2 - 6x - 4y + 9 = 0 v) z - 3 - 2i = 2 ) .

.

9

.

38. () z=x+yi, x, y R . ) , : x2 - i = y2 + 4 i)

y 2 1 0 1 2

()

3 4 x

ii) iii) iv) v) vi)

z- i = z- 4 z- 1- z- 4 = 015 2 Re (z) = 2Im (z) 8Re (z) = 15 + 2Im (z)

y = 4x -

) .

39. . , z1 = 3 + 4i, z2 = x + yi z3 = + i : ) 3 + 4 = 0. ) z2 z3.

y 4 3 2 1 0 1 2 3 4 x

40. z1 = 3 + i , z2 = 1 .

3 i.10

.

)

z1 = i. z2 az1 - z2 , w2006 = -1. z1 + az2

) w = ) z154 + z254 = 0 .

41. z =x+yi z1 = z- 2 + i z2 = 1+2i . z Im(z1z2) = 0. 42. z 8 Re z - = - Im( zi ) . z 43. z w z = 4w + Re(w) . A x2 y 2 + = 1 z 25 16 w x2 + y2 = 1. 44. z 1 y = 2x 1 , w = ( + i )z + 2 z + 3 . 45. i) N z z 2 - (4s unq) z + 4 = 0 , 0 < < , z C ii) (,) , . 46. z +

1 1 . = z , Re( z 2 ) = 2 z z1 z2

47. z1 , z2 C * z1 + z2 = z1 - z2 ,

.

.

11

.

48. z1 , z2 , z3 1 1 1 1 z1 + z2 + z3 = 7 , + + = . z1 z2 z3 7

z1 = z2 = z3 = 7

49. z 1 1. (1- z n )(1 + z ) f ( z) = . (1 + z n )(1- z )

1 i) f ( z ) = f , z 0 . z ii) z = 1 , f (z )= f ( z ) .50. f ( z ) =

z + 2i z C z - 2i . z - 2i

i) f ( z ) = 1 . ii) ( f ( z ))= f (z ) , z .51. i) z z - 2i + 3 = 2 w , w = -3z + i 5 .

ii) z = (2x-3)+(2y-1)i , x, y R , 2 z - 1 + 3i = 2 , (x,y) .52. z1 , z2 , z3 , z1 = z2 = z3 = r , r > 0 , , , , . w1=z1z2 , w2=z2z3 , w3=z3z1 , , . . 53. z , w , , . zw - iw = z - (2+3i) , (0,1) (2,3), x2+y2=1.

.

12

.

54. A) i) N z , (4+3i)z + (4-3i) z + 50 = 0 , 4x - 3y + 25 = 0 .

ii) ; ) i) z (z+3-3i) ( z +3+3i) = 4 . ii) z .55. z f ( z ) =

2 + z.i , z 1 . 1- z

i) f (2) = 2 2 . ii) f ( z) - 2 = z . f ( z) + i

iii) z = 1 f(z) , 3 y = -2x + . 256.

z - 1 = z - 3i , z C.

) z () (1, 0) (0, 3). ) () x - 3y + 4 = 0. ) (). ) z z .57. z f () = iz, * : f (4) + f (4 + 1) + f (4 + 2) + f (4 + 3) = 0, *.

1 1 58. z Re = , : z 4 ) z (2 , 0) = 2 , (0,0). . 13

.

) z Im(z) = 1, Re(z) = 2+ 3 Re(z) = 2- 3 .59. z w z z + i (z - z ) = 1 w () y = x + 1. : ) O z C

(0, 1) = 2 . ) () C () . ) t1, t2 () C, : t1 + t23n

+ t1 - t2

2n

= 23n + 1 .

60. ) (z) = z3 3z2 + 4z 12 . ) z3 3z2 + 4z 12 = 0 , z C . ) . ) ; . 6 1 . z 2 - 4e 2t z + 4e4t + e2t = 0 , z C , t R .

) . ) , . , . ) t . ( ).6 2 . z 2 - 8 z + l 2 + 9 = 0 , l R

z1 , z2 C .

) z1 = 5 l . ) = 4 , z1 = 4 + 3i , z2 = 4 - 3i .) z - z1 + z - z2 6 , z . . 14

.

- - 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23 24 25

-

1. ) x=4 , y= -2/3 , ) x = -2 , y= 7 , ) x =3 , y=2 , ) x = -1 2. ) z 2 = 77 36i . 3. ii) x=4 , y=2 4. ) x-y =4

99 99 1 1 1 1 1 1 1 1 1 + + ... + ) + i (1 + + + + + ...+ ) = .. 5. (1 + + 2 2 4 8 3 9 27 54 36. z1 = 4-6i , z2 = -2

.

15

.

7. ) 3/5 +11/5 i

8. ) 8-15i , ) 1/2 i , ) -

1 3 3 1 1 1 2 + i , ) + i , ) - + i 4 2 7 7 4 4

9. = 1, = 0 10. ( = 3, = - 2) ( = - 3, = 2)

11. x =

2 2

12. x = 1, y = 5 13. z = i, = - (2 - + 1) < 0 14. ) z =

2+i z=-

2+i 3 1 i 2 2) 1, -i , -1 , -3i

) z = 0 z = 1 z = -

=1 15. ) = 4, = 4 + 1, = 1 + i = 4 +2, = i = 4 +3, = 016. : 20n 20n (1+ i )(1+ i ) 1+ i = = ... = 1- i ( - i )( + i ) 1 1

5n 20n 2i = i4 = 1. 2

( )

18. ) 4x + 3y + 24 = 0 ) x2 + y2 + 6x + 8y = 0 ) (x + 3)2 + (y + 4)2 = 25 19. ) 2 + i ) = - 4, = 5

20. - 1, 1 + 2i, 1 - 2i

2 , ) =2-5 3 22. = 32 ..21. ) = 23. z1 = - 2i, z2 = 2 + 3i . 16

.

24. ) 25. ) 26. z =

2 2 1 23 +i 4

) ) 1

26

27. |z + 9|2 = 9 |z + 1|2 (z + 9) (z + 9) = 9 (z + 1) (z + 1)

(z + 9) ( z + 9) = 9 (z + 1) ( z + 1) ..z- 1 z- 1 = z+ 1 z+1

28. ) = -

(z - 1) ( z + 1) = - (z + 1) ( z - 1)

z z = 1 |z|2 = 1

|z| = 129. 3 + 4i, - 3 + 4i, 3 - 4i, - 3 - 4i (0, 0), R = 5

30. z = x + yi. x =

1 , y= 2

3 . 2

31. z = - 1 - i .

32. |1 - z| > |z|

|1 - z|2 > |z|2 1 2

(1 - z) (1 - z ) > z z

1 - z - z + zz > zz

z+ z 0 i= 0 i< 0

i2 > 0-1>0

i2 = 0- 1= 0

i2 > 0-1 >0

, . z . z = a + b i z = a - b i : i) z + z = 2a . . : . . z1 = 2 + 3i , z2 = 5 - 3i . ii) z - z = 2b i . . . z2 = 5 - i . . . z1 = 5 + 3i , iii) z = a2 + b 2 z

zR z= z zI z= - z: ( ). . 19

.

. . . .. z1 = 2 + i, z2 = 1 + 2i , z3 = 2 - i ... : i) z = z = - z

ii ) z = z z iii ) z1 2 = z1 z2 z iv) z n = zn

2

v) z = 0 z = 0 vi ) z1 - z2 z1 + z2 z1 + z2 . : i) z - z0 = r , r > 0 , A (z0 ) .

ii) z - z1 = z - z2 , A(z1 ), B (z2 ) . iii) z - z1 = r z - z2 , r 1, r > 0 , . iv) z - z1 + z - z2 = 2a, a > 0, z1 - z2 < 2a , . v) z - z1 - z - z2 = 2a, a > 0, z1 - z2 > 2a , . . ..i) z , w z2 + w2 = 0 , z=0 w=0 ( . z=1+i , w=1-i ) . ii) ( z + z ) 2 0 , ( z - z ) 2 0 , z + z = 2a R ,

z - z = 2b i I . .... . 20

.

z z . 1 : z = x + yi , x, y R , (x , y) . z x , y x y, z . 2 : : z = = ( ) , = (z ) z (0 , 0). z 1 z 2 = 1 M 2 = ( M 1 2 ) , M 1 = 1 (z 1 ) z 1

M 2 = M 2 (z 2 ) z 2 . ( ).

. 2 .

= z z o = , >0 z o = x + y i .

( ) = = (z )

= ( z ) z (x , y ) , . z .

(x x o ) 2 + (y y o ) 2 = 2

.

21

.

z z o , >0 z o = x + y i .

( ) = (z )

= ( z ) z (x , y ) , . . z .

(x x o ) 2 + (y y o ) 2 2

>

( ) > z z o > , >0 z o = x + y i . = (z )

= ( z ) z (x , y ) , ..

(x x o ) 2 + (y y o ) 2 > 2

z . . 22

.

uuuuu r uuuuu r

z z1 = z z 2 z 1 = x 1 + y 1 i z 2 = x 2 + y 2 i .

( ) = ( ) = (z ) , = (z 1 )

= ( z 2 )

z (x 1 , y1 ) z .

(x 2 , y 2 ) .

uuuuu r uuuuu r

z z1 z z 2 z 1 = x 1 + y 1 i z 2 = x 2 + y 2 i .

( ) ( ) = (z ) , = (z 1 ) = ( z 2 ) z (x 1 , y1 ) (x 2 , y 2 ) . 23

.

.

z ( , ) , .

+ = 2

z z 1 + z z 2 = 2 z 1 = + 0i

( ) + ( ) = 2 =( z ) , = (z 1 )

z 2 = + 0i

= ( z 2 )

2 > z 1 z 2 = 2 > 0

z ( , 0 ) ( , 0 ) .

x2 y 2 + =1 2 2

z , 2 .

.

24

.

+ = 2 z z 1 + z z 2 = 2 z 1 = 0 + ( )i z 2 = 0 + i

( ) + ( ) = 2 =( z ) , = (z 1 )

= ( z 2 ) z ( 0 , ) (0 , ) .

2 > z 1 z 2 = 2 > 0

x2 y 2 + =1 2 2

z , 2 .

=2 z z 1 z z 2 = 2 z 1 = + 0i

( ) ( ) =2 =( z ) , = (z 1 )

z 2 = + 0i

= ( z 2 )

.

25

.

2 = z 1 z 2 > 2 > 0

z ( , 0 ) ( , 0 ) .

x2 y2 =1 2 2

z , .

z z 1 z z 2 = 2 z 1 = + 0i

= 2

( ) ( ) = 2 =( z ) , = (z 1 )

z 2 = + 0i

= ( z 2 ) z ( , 0 ) ( , 0 ) , ( ) > ( ).

2 = z 1 z 2 > 2 > 0

x2 y 2 =1 , x >0 2 2

z , .

.

26

.

z z 1 z z 2 = 2 z 1 = 0 + ( )i z 2 = 0 + i

=2

( ) ( ) =2 =( z ) , = (z 1 )

= ( z 2 ) z ( 0 , ) (0 , ) .

2 = z 1 z 2 > 2 > 0

y2 2

x2 2

=1

z , .

= 2

z z 1 z z 2 = 2 z 1 = 0 + ( )i z 2 = 0 + i

( ) ( ) = 2 =( z ) , = (z 1 )

= ( z 2 ) z ( 0 , ) (0 , )

2 = z 1 z 2 > 2 > 0 .

y 2 x2 =1 , y >0 2 227

.

,

( ) > ( ). z , .

. : 1 z z 4 6i w = : z+4 ) )

z = x + y i , x , y R M (x , y). w z 4 (x , y) ( 4 , 0) . w ==

z 4 6i x + y i 4 6i (x 4) + (y 6) i [(x 4) + (y 6) i] [(x + 4) yi] = = = = [(x + 4) + yi] [(x + 4) yi] z+4 x + yi + 4 (x + 4) + yi

x 2 16 (x 4)yi + (x + 4)(y 6)i + (y 6)y x 2 + y 2 6y 16 + ( xy + 4y + xy 6x + 4y 24)i = = (x + 4) 2 + y 2 (x + 4) 2 + y 2

x 2 + y 2 6y 16 6x + 8y 24 = + i , (x , y) ( 4 , 0) . (x + 4) 2 + y 2 (x + 4) 2 + y 2

.

28

.

)

w x 2 + y 2 6y 16 = 0 x 2 + y 2 6y 16 =0 (x + 4) 2 + y 2 (x , y) ( 4 , 0)

x 2 + y 2 6y + 9 = 25 x 2 + (y 3) 2 = 25 (x , y) ( 4 , 0) (x , y) ( 4 , 0)( )

z ( 0 , 3 ) = 5, x 2 + (y 3) 2 = 25, ( 4 , 0) .

)

w 6x + 8y 24 = 0 6x + 8y 24 =0 (x + 4) 2 + y 2 (x , y) ( 4 , 0) 3x 4y + 12 = 0 (x , y) ( 4 , 0) z ( ) 3x 4y + 12 = 0 , ( 4 , 0) .

2 z w = ) z 1+ i : z +1+ i

)

z = x + y i , x , y R M (x , y). w z 1 i (x , y) (1 , 1) . w = z 1 + i x y i 1 + i (x 1) (y 1) i [(x 1) (y 1) i] [(x + 1) (y + 1)i] = = = = z + 1 + i x + y i + 1 + i (x + 1) + (y + 1)i [(x + 1) + (y + 1)i] [(x + 1) (y + 1)i]

=

x 2 1 (x 1)(y + 1)i (x + 1)(y 1)i (y 2 1) x 2 1 y 2 + 1 (xy + x y 1 + xy x + y 1)i = = (x + 1) 2 + (y + 1) 2 (x + 1) 2 + (y + 1) 2

.

29

.

2 (1 xy) x 2 y2 = + i , (x , y) (1 , 1) . 2 2 (x + 1) + (y + 1) (x + 1) 2 + (y + 1) 2

)

w x 2 y2 = 0 x 2 y2 =0 (x + 1) 2 + (y + 1) 2 (x , y) (1 , 1)

y=x (x y)(x + y) = 0 (x , y) (1 , 1) (x , y) (1 , 1) z , 1 : y = x 2 : y = x , (1 , 1).

)

w 1 xy= 0 2 (1 xy) =0 (x + 1) 2 + (y + 1) 2 (x , y) (1 , 1)1 xy =1 y= x (x , y) (1 , 1) (x , y) (1 , 1) z y =

( 1 , 1) .

1 , x

3 , , z , 2z + 3 z + 4i . z , : ) , , ) = 90

z = x + y i , x , y R (x , y). 2z + 3 = 2(x + yi) + 3 = (2x + 3) + 2yi (2x + 3 , 2y) , z + 4i = x + yi + 4i = x + (y + 4)i ( x , y + 4 ) . . 30

.

:

= (2x + 3 x , 2y y) = (x + 3 , y)

= (x 2x 3 , y + 4 2y) = ( x 3 , 4 y).

)

, , // det , = 0 x+3 y =0 (x + 3)(4 y) ( x 3) y = 0 x 3 4 y 4x xy + 12 3y + xy + 3y = 0 4x + 12 = 0

(

)

x = 3.

z : x = 3.

)

= 0

( x + 3 )( x 3 ) + y ( 4 y) = 0 x 2 6 x 9 + 4 y y 2 = 0 x 2 + y 2 + 6x 4 y + 9 = 0 ( 1 ) .

(1) x 2 + y 2 + x + y + = 0 = 6 , = 4 = 9. 2 + 2 4 = 36 + 16 36 = 16 > 0 , ( 1 ) , , ( 3 , 2 ) 2 2 =

1 2 + 2 4 = 2 2

z ( 3 , 2 ) = 2 , (x + 3) 2 + (y 2) 2 = 4.

: 1 z , z = ( + 1) + (2 1) i , R .

. 31

.

z = x + y i , x , y R M ( x , y) . x = + 1 (2) y = 2 1 2x + y = 3 y = 2x 3. 2x = 2 2 y = 2 1

z : y = 2 x 3.

2 z , z = 4 + i , R .

z = x + y i , x , y R x=4 M ( x , y) , y= M ( x , y) x = 4 y =

R. z : x = 4 .

3 z , z = 2 3 i , R .

z = x + y i , x , y R x = 2 M ( x , y) , y = 3 M ( x , y)

y = 3 x = 2 ,

. z

.

32

.

y = 3 : x0

4 z , z = 2 + i , R .

z = x + y i , x , y R x= 2 M ( x , y) , y =

M ( x , y) x = 2 y = ,

[ 1 , 1]. z , ( 2 , 1) ( 2 , 1) .

5 z , z = + i , R .

z = x + y i , x , y R x = M ( x , y) , y = R

2 + 2 = 1 x 2 + y 2 = 1 z ( 0 , 0 ) = 1 , x 2 + y 2 = 1. . 33

.

, , 2 2 ( 1 4 ), > 0 ,

M ( x , y) y = > 0 , .

z x 2 + y 2 = 1 ( 1 , 0 ) (1 , 0 ) .

6 z , z = ( 1 + ) + ( 2 ) i , R .

z = x + y i , x , y R x = 1 + M ( x , y) , y = 2 x 1 = y + 2 =

R 2 + 2 = 1 ( x 1) 2 + ( y + 2) 2 = 1 z (1 , 2 ) = 1, ( x 1) 2 + ( y + 2) 2 = 1.

7

.

34

.

z , z = 3 + 5 i , R .

z = x + y i , x , y R x = 3 M ( x , y) , y = 5

x = 3 y = 5 R 2 + 2 = 1 x 2 y2 + =1 9 25

z x 2 y2 + = 1. 9 25

8 1 z , z = + i , + , Z . 2

z = x + y i , x , y R 1 x= M ( x , y) , y =

+ 1 + 2 =

, Z 2

1 1+ y2 = x 2 x 2 y2 = 1 2

z x 2 y 2 = 1. 1 , , 2 2 . 35

.

( 1 4 ), > 0 , M ( x , y) x = .

1 > 0 ,

z x 2 y 2 = 1.

2

3 8 , 0 , U , 2 2 ( 1 3 ), 0 , M ( x , y) y = 0 , .

z x 2 y 2 = 1.

: 1 z z 2 + 3i = 5 .

z = x + y i , x , y R M ( x , y) . z 2 + 3i = 5 z ( 2 3i ) = 5 ( ) = 5 , = (z ) z ( 2 , 3 ). z ( 2 , 3 ) , 5 . z ( 2 , 3 ) = 5 , (x 2) 2 + (y + 3) 2 = 25. . 36

.

2 z z 2 + 3i 5 .

z = x + y i , x , y R M ( x , y) . z 2 + 3i 5 z ( 2 3i ) 5 ( ) 5 , = (z ) z ( 2 , 3 ). z ( 2 , 3 ), 5 . z ( 2 , 3 ) = 5 , (x 2) 2 + (y + 3) 2 25.

3 z z 2 + 3i > 5 .

z = x + y i , x , y R M ( x , y) . z 2 + 3i > 5 z (23i ) > 5 ( ) > 5 , = (z ) z ( 2 , 3). z ( 2 ,3). 5 . z (x 2) 2 + (y + 3) 2 = 25 ( 2 , 3 )

= 5.

4 z z + 2 + 8i = z 1 5i .

z = x + y i , x , y R M ( x , y) . . 37

.

z + 2 + 8i = z 1 5i z ( 2 8i ) = z (1 + 5i ) ( ) = ( ) , = (z ) z ,

( 2 , 8 ) (1 , 5 ). z ( 2 , 8 ) (1 , 5 ). z ( ) .

( ) : z + 2 + 8i = z 1 5i x + yi + 2 + 8i = x + yi1 5i (x + 2) + (y + 8)i = (x 1) + (y 5) i (x + 2) 2 + (y + 8) 2 = (x 1) 2 + (y 5) 2

x 2 + 4x + 4 + y 2 +16y + 64 = x 2 2x +1 + y 2 10y + 25 6 x + 26y + 42 = 0 3x + 13y + 21 = 0 .

5 z z + 2 + 8i z 1 5i .

z = x + y i , x , y R M ( x , y) . z + 2 + 8i z 1 5i z ( 2 8i ) z (1 + 5i )

( ) ( ) , = (z ) z ,

( 2 , 8 ) (1 , 5 ). z ( 2 , 8) (1 , 5 ). z ( , ) , ( ) . 3x + 13y + 21 0 .

.

38

.

6 z z + 4 + z 4 = 10 .

z = x + y i , x , y R M ( x , y) . z + 4 + z 4 = 10 z ( 4 + 0i ) + z ( 4 + 0i ) = 10

( ) + ( ) =10 , = (z ) z,

( 4 , 0 ) ( 4 , 0). z , 2 = 10 = 8 . z ( 4 , 0 ) ( 4 , 0), = 4 2 = 10 = 5, 2 = 2 2 2 = 25 16 2 = 9 = 3 , x 2 y2 x 2 y2 + 2 =1 + =1 25 9 2

7 z z + 8i + z 8i = 20 .

z = x + y i , x , y R M ( x , y) . z + 8i + z 8i = 20 z ( 0 8i ) + z ( 0 + 8i ) = 20

( ) + ( ) = 20 , = (z ) z,

(0 , 8 ) ( 0 , 8). z , 2 = 20 = 16 . . 39

.

z (0 , 8 ) (0 , 8) , = 8 2 = 20 = 10, 2 = 2 2 2 = 100 64 2 = 36 = 6 , x 2 y2 x 2 y2 + 2 =1 + =1 2 36 100

8 z z + 5 z 5 = 8.

z = x + y i , x , y R M ( x , y) . z + 5 z 5 = 8 z ( 5 + 0i ) z ( 5 + 0i ) = 8 ( ) ( ) = 8 , = (z ) z,

( 5 , 0 ) ( 5 , 0). z , 2 = 8 = 10 . z ( 5 , 0 ) ( 5 , 0 ) , = 5 , 2 = 8 = 4 , 2 = 2 2 2 = 25 16 2 = 9 = 3 , x 2 y2 x 2 y2 2 =1 =1 16 9 2

9 z z + 5 z 5 = 8 .

z = x + y i , x , y R M ( x , y) . . 40

.

z+5 z5 =8 z ( 5 + 0i ) z ( 5 + 0i ) = 8

( ) ( ) = 8 , = (z ) z, ( 5 , 0 ) ( 5 , 0). z , 2 = 8 , = 10 ( ) > ( ) . z (5 , 0 ) (5 , 0), = 5 2 = 8 = 4 , 2 = 2 2 2 = 25 16 2 = 9 = 3 , x 2 y2 x 2 y2 2 =1 =1 2 16 9

10 z z + 10 i z 10 i = 12 .

z = x + y i , x , y R M ( x , y) . z + 10i z 10i = 12

z ( 0 10i ) z ( 0 + 10i ) = 12 ( ) ( ) =12 , = (z ) z,

( 0 , 10 ) ( 0 , 10). z , 2 = 12 = 20 . z ( 0 , 10 ) (0 , 10 ) , = 10 , 2 = 12 = 6 , 2 = 2 2 2 = 100 36 2 = 64 = 8 , y2 x 2 y2 x 2 2 =1 =1 36 64 2

.

41

.

11 z z + 10i z 10i = 12 .

z = x + y i , x , y R M ( x , y) . z + 10i z 10i = 12 z ( 0 10i ) z ( 0 + 10i ) = 12 ( ) ( ) =12 , = (z ) z,

( 0 , 10 ) ( 0 , 10). z , 2 = 12 , = 20 ( ) > ( ) . z (0 , 10) (0 , 10), =10 2 =12 = 6 , 2 = 2 2 2 =100 36 2 = 64 = 8 , y2 x 2 y2 x 2 2 =1 =1 2 36 64

.

42

.

1 z1 = ) : i) 1 + z1 + z12 = 0 ii) z13 = 12 iii) z2 n + 1 = - z1n + 2 , 2 z2 n = z1n

1 3 + i z2 = 1 + z1 2 2

(n N * )

36 ) + i z2 z19 . 2

: 1 1 3 1 3 3 1 z = - + i = - 2 i - = - 2 2 2 2 4 2 4 1 3 1 3 (i)1 + z1 + z12 = 1 - + i- i= 0 2 2 2 22 1 2

3 i. 2

(ii)1 + z1 + z12 = 0 (z1 - 1)(1 + z1 + z12 )= 0 z13 - 1 = 0 z13 = 1 . (iii) 1 + z1 = - z12 z2 = - z12 . :2 z2 n + 1 = (- z12 ) 2n + 1

= - z14n + 2 = - z1n + 2 13n = - z1n + 2 (z13 ) = - z1n + 2 3 = - z1n+ 2 z 1

n

2 z2 n = (- z12 ) = z14n = z1n 13n = z1n = z1n . z 1

2n

) (iii) :36 18 z2 = z1 = (z13 ) = 16 = 1 + 0i 6

29 z19 = z2+ 1 = - z19+ 2 = - z12 19 = - z12 = - z12 = z 1 2

1 3 + i. 2 2

.

43

.

2 z = x + yi , x,y R . 2z-iz f z = , z 1. z-1

()

)

f (1- i) = 3 + 3i.2004

1 ) (f ( - i ))

.

) , f (1-i ) f (1+i ) . . ( ) :) f (1 - i )= ) 2(1 - i )- i ( + i ) 3 - 3i 1 = = 3 + 3i -i 1- i - 12004 2004

( f (1 - i))

2004

= (3 + 3i )1002

= (3( + i )) 1

=

2 1002 32004 ( + i ) = 32004 (2i ) = 21002 2004 R 3 1

2( + i )- i ( - i ) 2 + 2i - i + i 2 1 + i 1 1 = = = 1- i 1 ) f ( + i )= 1+ i - 1 i i uuu uuu r r ( 3 , 3 ) , (1 ,- 1 ) OA = (3,3)( - 1)= 3 - 3 = 0 OB 1,

O = 900 .

3 z f (n )= i n , *. z ) : f (1)+f (6)+f (7 )+f (16)=0 ) : f (n ) + f (n + 2) + f (n + 4)+ f (n + 6)= 0 ) |z| = 2 f (2001)+f (2004) =2 2 . . 44

.

: ) f (1)+ f (6)+ f (7)+ f ( )= i + i 6 + i 7 + i16 = 16 z z z z

iz + i 2 z + i 3 z + z = iz - z - iz + z = 0 ) f (n )+ f (n + 2)+ f (n + 4)+ f (n + 6)=in + in+ 2 + in+ 4 + in+ 6 = z z z z in z - in z + in z - in z = 0 ) f (2001)+ f (2004) = i 2001 z + i 2004 z = iz + z = z ( + i) = z 2 = 2 2 1

4 z = x + yi , x,y R ) z - (Im (z )+ 1) + 1 = 0 , z 1 y = x 2 . 2 ) z () 8 . ) z , () |z| = . : )z - (Im (z )+ 1) + 1 = 0 2 22 2

(

x2 + y 2

) - (y + 1) + 1 = 0 2

2

x2 + y 2 - y 2 - 2 y - 1 + 1 = 0 x2 - 2 y = 0 1 y = x2 (I ) 2 ) :

.

45

.

( 8) -

2

(y + 1) + 1 = 0 8 - (y + 1) + 1 = 0 2

2

2

(y + 1) = 9 y + 1 = 3 y = 2 y = - 4 () z2 = - 2 + 2i z1 = 2 + 2i, ) 2 r 2 - (y + 1) + 1 = 0 r 2 - y 2 - 2 y - 1 + 1 = 0 y2 + 2 y - r 2 = 0 D = 4 + 4r 2 > 0 y1 2 = - r 2 < 0 y () , .

5 ) () z : |z| = 2 Re (z ) 0 . ) z () w = z 4 + 3i . ) w . :) z = 2 Re( z ) 0 yy (0,2) (0, -2) (2 , 0) . ) w = a + b i z = x + yi w = z - 4 + 3i z = w + 4 - 3i x + yi = (a + 4)+ (b - 3)i 0 x 2 - 2 y 2 : 0 a + 4 2 - 4 a - 2 - 2 b - 3 2 1 b 5 ( I ) z = w + 4 - 3i z = w + 4 - 3i w + 4 - 3i = 2 . w (-4 , 3) = 2. - 4 a - 2 1 b 5 . w = - 4. . 46

.

) y = l x (-4 , 3 ) 3 , 3 = l (- 4) l = - . 4 3 y = - x 4 2 2 (x + 4) + (y - 3) = 4 , - 4 x - 2 .

6 z w .uuu uuu r r ) Re (z )= OA . w OB

( ) .

) xx z-1, z-i. ) z, z-1, z-i , . . :) z = x + yi w = a + b i A(x, y ) B (a, b ) uuu uuu r r OA = (x, y )a, b )= ax + b y (I) OB ( z = (x + yi )(a - b i )= ax - b xi + ayi + b y = (ax + b y )+ (ay - b x)i (II) w uuu uuu r r () () Re (z )= OA . w OB ) z - 1 = (x - 1)+ yi G(x - 1, y ). z - i = x + (y - 1)i D (x, y - 1). l GD = y - 1- y = - 1 1350. x- x+ 1

) A(x, y ) , G(x - 1, y ), D (x, y - 1) // // yy ^ .

.

47

.

7 z : z 6 = (z + 1) =1 . ) : i) z =1 ii) z 2 + z +1= 0 . (I) z6

) () z3 = 1. . :) i ) z 6 = 1 z = 1 z = 1 z = 1 z = z6 6 2 6 2

1 . z

ii ) (z + 1) = 1 z + 1 = 1 z + 1 = 1 (z + 1)(z + 1)= 1 z z + z + z + 1= 1 1+ z + z = 0 1+ z + 1 = 0 z

z + z2 + 1 = 0 z 2 + z + 1 = 0. z ) z 2 + z + 1 = 0 1 3 1 3 z1 = - + i , z2 = - - i , 2 2 2 2 1 3 1 3 , : A( ) , B - , , G- , 1,0 2 2 2 2 () = () = () = 3 .

8 z1 , z2 : | z2 | = 1 | z1 z2 | = | z1 | . ) A Re( z1 2 ) = z1 2

) , : z2 = z1 ( - 2l i ). 1 . 48

.

) l =

1 2 z1 = . 2 2

) z1 z2 , . ( ). :) z1 - z2 = z1 z1 - z2 = z1 (z1 - z2 )(z1 - z2 )= z1 z1 2 2

z1 z1 - z1 z2 - z2 z1 + z2 z2 = z1 z1 z1 z2 + z1 z2 = 1 2Re (z1 z2 )= 1 Re(z1 z2 )= 1 2

l l ) z2 = z1 (1 - 2l i ) z2 = z1 - 2 z1 i z1 - z2 = 2 z1 i 1 1 l = , (l > 0). z1 - z2 = 2 z1l i z1 = 2 z1 l i l = 2 2 1 2 ) z2 = z1 ( - 2l i ) l = 1 z2 = z1 ( - i ) z2 = z1 - i z1 = 1 . 1 2 2 2 2 2 (AB )= . 2 2 (OA)= (OB )= 1 . 2 (OA)= (AB ) . ) z1 - z2 = z1 =

.

49

.

1) () z : |z| = 3 Im (z ) 0 . ) z () 1 9 = z + 2 z .

2 z=+i , R = 2 z + i z - 3 z z. ) Re() = 2 + - 3 Im()= + 2 . ) y = 3x + 2, z y = - 5x + 7. ) z y = - 5x + 7 .

3) z |z- 3- i | 2. ) : z - (3+i ) - (3-i)= z - 6 . ) : z - 6 2 + 10

) z1 , z2 () z1 - z2 4 . . 50

.

4 z w : zz + z + z = 3 , w = l - (l + 1)i , R . ) z (C) (-1 , 0) = 2. ) w () : + y + 1 = 0. ) () (C) . ) z . ) w .

5z z 0 f (v)= (i n - 1) , *.

) * : f (n )f (n + 1)f (n + 2)f (n + 3)= 0 . ) f (5)= - 3 + i , z = 2 + i. ) z = 2 + i. f (n + 3)- f (n + 1) = 2 5 , *.

6

z . z2 + 1 ) w , z |z| = 1 . z 3 ) , : 2 = . z +1 3 z w =

.

51

.

) z1 , z2 () 2005 z (z1 2 ) - i . : K = 2004 z1 + z 2 4+ 3

7 z1 , z2 , z3 | z1| = | z2 | = | z3 | = 3 . 9 ) z1 = . z1 z z ) 1 + 2 . z2 z1 1 ) z1 + z 2 + z3 = z1z 2 + z 2 z3 + z3z1 . 3 9 ) z1 = z 2 = z1 - z 2 Re (z1 z2 ) = . 2

8 z 0 , : z = 1 - z = () ) z + z = 1 . ) () 1 3 1 3 z1 = + i , z2 = - i . 2 2 2 2 ) , z1 , z2 , AOB , ( ). ) w = 1 z .

(z1 + z2 )n z1n + z2

n

.

.

52

.

9 z w = z+i z+1

) z z = - z . ) |w| = 1 : Re (z ) = Im (z ). ) w , z y = x y = -x-1. ) |w| = 2 z .

10 z = x + yi w = 1- z , z 0 . : z

w = 2 . w) z

(x

+ 1) + y 2 = 2 .

2

) z . ) u = w+ 1 .

11 f (z )= 1= z - i 2 = f (z)- i . iz + 2 - i , z i. z- i

) 12 = 1 - i. ) z x 2 + y 2 = 4 . 53

.

i) 1 (0 , - 1 ) = 2. ii) 2 . .

12

3 5 i w = 2 + iz - z , z = x + yi , x,y R . 2 2 ) Re (w )= 2 (x-2y ) Im (w )= - (x - 2y). ) w y= ) w u =5-i . ) w = x-2y 5 . ) . . z = x + yi 1 w = 5 . 21 x. 2

13 z, w u = z : z - w = z + w . w ) z z = -z . ) u , w1 = z . w ) w= 2 - i, z yy . ) () w; ( ).

.

54

.

14 z1= 1 2i z2 = 3 +4i . z ) 2 = x + yi , x , y R x = -1 y = 2 . z1

) x2 + x+2 = 0 , , R .

z2 , z1

) z z - 2 z1 = z2 .

15o f ) f (z ) = f (z )=z + i , z z 0 . z

f (z ) , z .

) f (z ) = 1 , . . z . ) Re ( f (z )) = 2 , z , .

16 z : 1 1 4 + = 2 . z- i z + i z +1

) .. z . ) 1, 2 z1 , z2 . . , z2 - z1 .

.

55

.

17 z : z - i + 3 =5 . ) . . = z -1+2i. ) || . ) .

18 z = x + yi , x, y R ; 1 4 z - 4Im (z)+ + 5 = 0 . 22 2

) z y = x2 + 1. ) . * ) , z1 = 1 + 2i z2 = - 1 + 2i . y = x2 + 1 2/3 ..

19o z = + i , , R : | z | = 1. z f (x )= x + z , x R . ) f (x )= x 2 + 2Re(z 2 ) + 1 . x ) z f xx ;2

.

56

.

) w =

1 + 2i , .. w . z

) z , w .

20 z , z 0 : z-i = 2 . z

) z ( 0 , - 1) R = 2 . ) z . ) w : w = z 2i . . . w .

21 z , w , z 0 ,w 0 OM OM 2 1 , . ) uuuuuur uuuuuu r u OM +OM . 1 2uuuuu r

uuuuuu r

uuuuuu r

OM z , w

) : | z + w | = | z w |. : i)

OM1 ^ OM 2

uuuuu r

uuuuu r

ii) | z w |2 = | z |2 + | w |2

w iii) Re (z ) = 0 .

.

57

.

22 z : | z - 2 + i | = 1 . ) z ( 2 , - 1 ) R = 1 . ) z1 , z2 , : | z1 z2 | 2 . ) w = z + 2 i w .

) | w - z | . ) | z | .

23 z1 , z2 , z3 : z1 = z2 = z3 = 1 z1 + z2 + z3 = 0 . : ) z1 + z2 + z1 - z2 = 2 z1 + 2 z2 ) z1 - z2 = z1 - z3 = z2 - z3 ) z1 - z2 = 3 Re(z1 z2 )= 2 2 2 2 2

1 . 2

) z1 , z2 , z3 .

24 z = x+yi . 3 - iz , z - 1 . f ( z) = 1+ z . 58

.

) ( f (3)) . ) : i)f ( z) - 3 = z. f ( z) + i

10

ii) z = 1 , f(z) 3x+y-4=0 . ) f(z) (ii) , .

.

59

.

-

1 ) z = 3 z , (0,0) =3. Im( z ) 0 y 0 . z = x + yi xx.

=x x+ () -3 x 3 . yi +

) z = x + yi w =

1

x + 2

9

yi

2

) z =

35 26

+

7 26

i.

3

) (3,1) =2. ) .

) z1 - z 2 z1 , z 2 . 4 ) () (-1,0). ) z = - 3 )

{ x+ y+ 1= 0

y= x

-

1 2

,-

1

2.

5

)

2 n 2 n+ 1 i ) - 1 z i ) - 1 z ( (

.)

f (n + 3) - f (n + 1) =

(in+3- 1)z- (in+1- 1)z = 2 in+1z = 2 i n+1 2+ i = 2 5

.

6

) z1 =

3+ i 2

, z2 =

3- i 23 , K = 1- i 560

) Vieta z1. z 2 = 1 k a i z1 + z 2 = .

.

7

) z1 + z 2 + z3 = z1 + z 2 + z3 = z1 + z 2 + z3 =

9 z1

+

9 z2

+

9 z3

( z1z 2 z3 = z1 z 2 z3 = 27 ) .) z1 = x + yi z 2 = a + b i Re( z1 z2 ) = a x + b y . 2 z1 - z 2 = 3 z1 - z 2 = 9 ...

8

) z = 1 - z z

2

= 1- z

2

z z = (1 - z )(1 - z ) z + z = 1 .

) z =

1 z

z =

1 z

() z +

1 z

2 = 1 z - z + 1 = 0.

. .

)

uur 1 OA = , 2

3 2

,

uur 1 OB = , 2

3 2

s un AOB =

( )

OA OB

uur uur uur uurOA.OB

= -

1 2

,

o AOB = 120 .) w = w .

9

) w = 1

z+ i z+1

= 1 z + i = z + 1

(z + i)(z -

i = z + 1 (z + 1) i z - z = z + z x = y) w I w = - w

) (

)

(

)

..

- 4 1 2 2 ) w = 2 ww = 4 .. . K , R = . 3 3 3 10 ) z1 = ) u =1 2 - 1 . z 2 = 2 - 1.

z

. u

1 z2

.

61

.

u

1 z1

.

110

) w2 =

1- i z- i

.

) i) E z = w1 + i z = 2 w1 + i = 2 ii) w2 =

1- i w1

w2 =

1- i w1

w2 =

2w1

. w2

w1 w2 w1 . () w2 22 5 11 5max

= 2 , w2

min

=

2 3

.

12

) w =

-

i x - 2y = 2 .

) x - 2 y = - 2

13

) u I u = - u zw = - zw zw = - z w w1 = - w1 w1 I .

) z (2,1) (-2,1) y. ) , z - w = z + w , - w w

y.

) x1 = - 1 + 2i , x2 = - 1 - 2i vieta =2 =5/2.z - 2 z1 = z 2 z - 2 ( - 2i) = 3 + 4i z - 2 + 4i = 5 , 1 z (2,-4) R=5.

14

)

.

15

)

f ( z) = f ( z)

z+ i z

=

z+ i z

2i z = 2i z z = z z R.) y = .

- 1 2

.62

.

2 2 ) x + y - y = 0 , (0,1/2) R=1/2.: () . z (0,0).

16

)

1

z- i

+

1

z+ i

=

4 2 z +1

1 z- i

+

1 z+ i

=

4 2 2 z - i

z + i + z - i = 4 . z (0,-1) (0,1) . . z , =2.) min z 2 - z1 = 2b = 2 3

max z 2 - z1 = 2a = 4 . 17 ) w = z - 1 + 2i = z - i + 3 + i - 3 - 1 + 2i = z - i + 3 - 4 + 3i .

w + 4 - 3i = z - i + 3 . w + 4 - 3i = 5 .

w+ 4- 3i = z- i + 3

. (-4,3) R=5.) .

max w = 10 .) =0 . 18 ) z = i

2 1 2 ) : E = 2 0 x + 1 - 2 x dx = ... = t .m. 3

(

)

19

) f ( x ) = xz + z

(

)(x z + z)= ... = x 2 + x (z 2 + z 2 )+ 1 ==

2 2 2 2 2 = x + 2 a - b x + 1 = x + 2 Re z x + 1 .) f ( x ) = 0 . =0

(

)

( )

.

63

.

4 Re z z = 1 a 2 + b 2

(

2

)

2

2 2 2 2 4 = 0 a - b = - 1 a - b = 1 z = 1 z = -1 z = i

= 1 =1 =-1 =1 =-1 .

z = -i. w - 2i = 1. z z z w (0,2) R=1.) . 20 ) : ) w =

1

+ 2i w - 2i =

1

w - 2i =

1

z- i =

2 2 2 z ... z z + i z - z - 1 = 0 ... x + (y + 1) = 2 .

(

)

) z =

( 2 - 1)i .( 2 + 1)i .

z = ) w = z - 2i w + 2i = z

(1)

z + i = w+ 3i = z + i

2

( () ) (1) : w + 3i = z + i 2 . . w

w + 3i =

(0,-3) = 2 . 21 ) .

) i) z + w = z - w

OM1.OM 2 = 0 OM1 ^ OM 2 .

uuuu uuuur r

OM = M1M 2 OM1 + OM 2 = OM1 - OM 2 ,

uuu r

uuuuuu r

uuuu r

uuuur

uuuu uuuur r

uuuu r

uuuur

ii) ( ) z w + z w = 0

z- w

2

= (z - w) z - w = ... = z

(

)

2

2 + w -

(zw + zw)=

z

2

2 + w .

iii) z w + z w = 0 ,

.: . . 64

.

22

) .

z1 - z2 2 ( 2 () ).) w = z + 2i w - 2i = z w - 2i - 2 + i = z - 2 + i ,

w- 2- i = z - 2+ i w-

(2 + i) = 1 .

(2,1) =1.

) max w - z = 4 . ) min z =max z = 5- 1 5 + 1.

23

) . 9 . 101 . . ) () ( )

- z3

2

+ z1 - z 2

2

= 2 z1

2

+ 2 z2

2

1 + z1 - z2

2

= 2 + 2 z1 - z 2

2

= 3 .

z1 - z3 = z 2 - z3 = 3 .) z1 - z 2 Re z1 z 2 =

2

= 3

(z1 -

z2

)(z1 -

z 2 = 3 ... 2 (a x + b y )= 1

)

( )

1 2

. ( z1 = x + yi , z 2 = a + b i ).

) () . 2 2 x + y = 1 .

: ) () ( ()).z1 - z 2 = z1 - z3 - z 2 - z3 - z 2 = - z 2 - z3 - z3 z3 + 2 z 2 = z 2 + 2 z3 ...

) .

24

)

( f (3))

10

= - 2

- 15 10

3 i

) z =

6 5

+

2 5

i .

.

65

.

.

= 1 + ( 1) = (1 + ) 2

= 1 11 ( 1) , 1 = 1

u ur u ur r r : . = ,

.u ur r . . = x1 x2 + y1 y2 , u r ur = ( x1 , y1 ) = ( x2 , y2 ) . : A( x1 , y1 ) B( x2 , y2 ) d ( , ) = ( x2 x1 ) 2 + ( y2 y1 ) 2

: A( x0 , y0 ) ( ) : x + y + = 0 x0 + y0 + d ( , ) = 2 + 2 : ( ) : x + y + = 0 , = . 1 = 2 1.2 = 1 . 1) (0,0) .x2 + y 2 = 2

( x0 , y0 ) . ( x x0 ) 2 + ( y y0 ) 2 = 2

.

66

.

x 2 + y 2 + Ax + By + = 0 K (

A B , ) 2 2

A2 + B 2 4 A2 + B 2 4 > 0 R = 2 . p p 2) : y 2 = 2 px , E ( ,0) x = 2 2 p p x 2 = 2 py , E (0, ) y = 2 2

3) E:

x2

2 2 2 =2 2.x2

+

y2

= 1 , ( ,0) ( ,0) ,

2 2 2 =2 2.x2 y2

+

y2

= 1 , (0, ) (0, ) ,

4) :

2 2 2 = 2 2.y2

= 1 , ( ,0) ( ,0) ,

2 2 2 = 2 2.

x2

= 1 , (0, ) (0, ) ,

.

67

.

1) , .

( a + i ). ( z + z = 2a . . . 91). z R z = z( . .8 97).

(: ).

xx .2) , .

( a + i ). ( z z = 2 i . . . 91). z I z = z( . .8 97).

(: ).

yy .

3) i 4 ( i 4 = 1). 4) , , 2 z = z z . . 68

.

5) . ) . (). ( ). i) (0,0) x0 + y0 + . = (), d (O, ) = 2 + 2 2 + 2( ).

ii) , () () . ( y = x , . = 1 ..). ) . R. ( ).

i) z .. :z min = (OK ) R = 13 2

z max = (OK ) + R = 13 + 2 .

( ()

(OK ) = ( x2 x1 ) 2 + ( y2 y1 ) 2

.

ii) . . 69

.

y = x , ...) . 2 y = x . 3 ( x + 3) 2 + ( y 2) 2 = 4 (

6) ( ) z1 , z2 , z3 , z1 z2 ...

.

70