ملزمة رياضيات سادس علمي _ العراق
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-
2015 / 2016
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80087430770
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2015 / 2016
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80087430770
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) ( [ 1 1 ]
80087450770 / 5
: ( )
: 2x 0 = 16+ : ]1 [1
x2 +16 = 0 x2 = -16 x = 16 = 16. 1 = 41
. (1-) 1 , 2x + 16 = 0
2x + 16 = 0 ( Complex Number )
(Imaginary Numbers) i = 1 { 4i} 2x + 16 = 0
.
i :- i = 1 i2 = -1
i3 = i2 . i = -1 . i = -i
i4 = i2 . i2 = (-1) . (-1) = 1 i4 = 1
:
i4n + r = ir , n N . r = 0 , 1 , 2 , 3 {i , -1 , -i , 1} i
i 4i:
i = i . 6i = 1 . 6)4i = (i . 24= i 25i /
i99= i96 . i3 =(i4)24 . i3 = 124 . i3 = i3 = -i
- : /1
i20 = i24 . i3 = (i4)6 . i3 = 16 . i3 = -i
i08 = i08 . i = (i4)28
. i = 128 . i = i
i0 = i4 . i3 = 8 . i3 = -i
i81 = (i4)4 = 8
i30 = i31 . i2 =(i4)84
. i2 =184 (-1)= -1
i104 = (i4)26
= 126 = 1
i10 = i8 . i2 = (i4)2 . i2 = 8 . i2 = -1
i17 = i16 . i = (i4)4 . i = 14 . i = i
i12n+93 = i12n . i93 = (i4)3n . (i4)23 . i = 13n . 1 . i = i
)ComplexNumber(, = 1i a,b c = a + bi :. Imaginary Part b Real Part a
. a+bi
-
) ( [ 1 1 ]
80087450770 / 6
i -13 = i -13 . (i4)4 =i -13 .(i16)= i3 = -i OR i -13 = 1
i131 =
i16
i131 = i3 = -i
( i ) (1) i / .(i) (4)
: i /
16 = 16 .1 = 4 i
25 = 25 .1 = 5 i
12 = 12 .1 = 23 i
15 = 15 .1 = 15 i :
a = a .1 = a i , a 0
, : (a , b) c = a + bi/
2 + 3i = (2,3) -1 + i = (-1 , 1) 2 = 2 + 0 i = (2,0) 3i = 0 + 3 i = (0,3)
:a+bi /2
a) 100 = 100 .i = 0 +10i
b) -1 + 3 = -1 + 3 i
c) 1+25
4 =
1
4 +
25
4 =
1
4 +
5i
4
d) -5 = -5 + 0 i
, (a,0) a + 0i a :
: 1]- [2 :/
9i 32 3i)+(7 - 12i) = 32 9i + 25) : ) (
: /3
a) 3 + 4 2 i , 5 - 22 i b) 3 , 2 5 i c) 1 3 i , i
/
a) (3 + 4 2 i) + (5 - 22 i) = 8 + 22 i
. R C /
- :
i2+ b 1) + (b2+ a 1(a= 2c+ 1c( : 2,c1c i1b + 1= a 1c ,i2b + 2= a 2c . R )2+ b 1(b ,R )2+ a 1(a :
(a1 + a2) + (b1 + b2)i
-
) ( [ 1 1 ]
80087450770 / 0
b) )3 + 0i) + (2 5 i) = 5 5 i c) (1 3i) + (0 + i) = 1 2i
3, c 2, c 1c : Commutativity): 1+ c2 c =2 c +1 c ) (1 Associativity): 3 ) + c2+ c1 (c = )3 + c2 (c +1 c ) (2
Additive Inverse): -c c = a + bi , c ) (3 .c (c-) c = -a-bi
e = 0 = 0 + 0i e :(AdditiveIdentity ) (4 Commutative Group (+ , )
:/4a) (4-5i) - (3+2i) = (4-5i) + (-3-2i) = 1-7i b) (7-13i) - (9+4i) = (7-13i) + (-9-4i) = -2-17i
,x (2-4i)+x = 5+i /5 /
(2-4i)+x = 5+i x = (5+i) (2-4i) = (5+i) +(-2+4i) x = 3 + 5i
: /
: )-(1 2i
: i1b + 1= a 1c ,i2b + 2= a 2c
c1. c2 = (a1 +b1i) (a2 +b2i)= a1 .a2 + a1 .b2i + a2 .b1i + b1 .b2i2 = a1 .a2 + a1 .b2i + a2 .b1i - b1 .b2 = (a1 .a2 - b1 .b2) + (a1 .b2 + a2 .b1)i
/
(2+5i).(3-4i) = 6 8i + 15i 20 i2 i2 = -1 = 6 + 20 + 7i = 26 + 7i
) 26 + 7i (
/ .
kc = ka + kbi: k R ,c = a + bi /
- :
: 2,c1c i1+b 1= a 1c ,i2+b 2= a 2c c1.c2 =(a1.a2 - b1.b2)+(a1.b2 + a2.b1)i
R )R )2b.1b - 2a.1a ( ,R )1b.2a+ 2b.1a :
. 2c.1c
-
) ( [ 1 1 ]
80087450770 / 0
3, c 2, c 1c : Commutativity): 1 cX2 c =2 c X1c ) (1 Associativity): 3c )2 c.1 c) = )3 c .2 (c .1 c ) (2 (0i+1)=1 (Multiplicative Identity ) (3
, (Multiplicative Inverse :) c (0+0i) (41
c
1
c = (1+0i) c x
c 1
c .
( - (0+0i), X) : (,+,X) :
: /6
1) (3+4i)2 2) i(1+i)
3) 5
2(4+3i)
4) (1+i)2 + (1-i)2 5) (1+i)3 + (1-i)3
/
1) (3+4i)2 = 9 + 24i 16 = -7 + 24i / .
2) i(1+i) = i + i2 = -1 + i
.(i+1) i/
3) 5
2(4+3i) = -10 -
15
2 i
4) (1+i)2 + (1-i)2 = (1+2i-1) +(12i-1) = 2i 2i = 0 ) (. /
5) (1+i)3 + (1-i)3 = (1+i)2 (1+i) + (1-i)2 (1-i)=2i(1+i) + (-2i)(1-i)= 2i - 2 - 2i 2 = -4 . /
: c=a+bi k R ,c /1) k. c = k)a + bi( = ka + kbi 2) ki. c = ki(a + bi) = -kb + kai
-
) ( [ 3 1 ]
80087450770 / 7
: 1] - 3[
4i+5 4i-5 , (i-) (i) , i-3 i+3 : . 0 0 ,
/ :1
1) = 1 2 2) . = .
3) (
)
= (
)
4) c = a2 + b2 c = a+bi 5) = c 6) c + = 2a 7) = c
/ :2
c1 c2 = c1 .1
c2 , c1 , c2
2c 0 2c c1 / 3
:
c1
c2 =
c1
c2 (
c2
c2 )=
c1.c2
c2.c2 =
c1.c2
a2+b2
: )( : 2+3i
45i
: (5i+4) /2+3i
45i =
2+3i
45i.4+5i
4+5i =
(2+3i)(4+5i)
42+52 =
8+12i+10i15
16+25 =
7+22i
41 =
7
41+
22i
41
: 2i-= 3 2= 1 + i , c 1c /7
a) c1 + c2 = c1 + c2 b) c1 .c2 = c1 . c2
c) (c1
c2)
= (
c1
c2 )
/
a) c1 + c2 = c1 + c2 c1 + c2 = (1+ i) + (3 - 2i) = 4 i c1 + c2 = 4 + i c1 + c2 = (1 i) + (3 + 2i) = 4 + i + = 1 + 2
b) c1. c2 = c1 . c2 c1. c2 = (1+ i) (3 - 2i) = 3 + 3i 2i + 2 = 5 + i c1. c2 = 5 i c1 . c2 = (1- i) (3 + 2i) = 3 - 3i + 2i + 2 = 5 i . = 1 . 2
c = a - bi R , a , b c = a + bi - :
/ )(
.
-
) ( [ 3 1 ]
80087450770 / 08
c) (c1
c2)
= (
c1
c2 )
c1
c2 =
1+ i
3 2i =
(1+ i)(3+ 2i)
9+4 =
3+3i+2i2
13 =
1+5i
13 =
1
13 +
5i
13 (
c1
c2)
=
1
13 -
5i
13
c1
c2 =
1 i
3+ 2i =
(1 i)(3 2i)
9+4 =
33i2i2
13 =
15i
13 =
1
13 -
5i
13 (
)
= (
)
. 2i-2 /8
2i-2 /1
2 2i
1
2 2i =
1
2 2i.2+ 2i
2+ 2i =
2+ 2i
4+4 =
2
8+
2i
8 =
1
4+
1
4i
)-2i ( y 2x +2 : )(
.x2 + y2 = x2 - y2 i2 = (x - yi)(x + yi)
. a,b a+bi 0.65, 10 , 53 /9 /
10 = 9 + 1 = 32 + 12 = 32 - 12 i2 = (3 i)(3+i) (i2-)
53 = 4 + 49 = 22 + 72 = 22 - 72 i2 = (27i) (2+7i) 0.65 = 0.49 + 0.16 = 0.49 - 0.16 i2 = (0.70.4i) (0.7+0.47i)
.
: y x /11
1) )2x -1( + 2i = 1 + (y+1)i 2x -1 = 1 2x = 2 x = 1 2 = y+1 y = 1
.) )
)(.
.
2) 3x 4i = 2 + 8yi
3x = 2 x = 2
3
-4 = 8y y = 1
2
3) (2y+1) (2x-1)i = -8 + 3i
2y+1 = - 8 2y = -9 y = 9
2
-(2x-1) = 3 -2x = 2 x = -1
2 + 2i
b 1, b 2= a 1a 2c =1c =2 : i1+b 1= a 1c ,i2+b 2= a 2c : - :
-
) ( [ 3 1 ]
80087450770 / 00
4) (x + yi)(3 + 2i) = 5 - 3i
(x + yi) = 5 3i
3 + 2i =
5 3i
3 + 2i .
3 2i
3 2i
(x + yi) = 159i10i6
9+4 =
919i
13 =
9
13
19
13 i
x = 9
13 , y = 19
13
5) xyi
(3+i)2 = 1 2i x yi = (1 2i)(3 + i)2
= (1 2i)(9 + 6i 1) = (1 2i)(8 + 6i) = (8 + 6i 16i 12 i2) = (20 10 i)
x = 20 & y = 10
6) 32i
i ,
xyi
1+5i
= c xyi
1+5i
c = xyi
1+5i c =
x+yi
15i (
c1
c2)
= (
c1
c2 )
x+yi15i
= 32i
i
(x + yi)i = (1 5i)(3 2i) xi y = 3 15i 2i 10 y + xi = -7 17i
-y = -7 y = 7 x = -17
1 1
n (8 N: 1. i5 = i4 . i = i = 0 + i 2. i6 = i4 . i2 = -1 = -1 + 0i 3. i124 = (i4)31 = 1 = 1 + 0i 4. i999 = i996. i3 =(i4)247 . i3 = -i = 0 - i 5. i4n+1 = i4n . i = i = 0 + i
6. (2 + 3i)2 + (12 + 2i) = (4 + 12i - 9) + (12 + 2i) = (-5 + 12i) + (12 + 2i) = 7 + 14i
7. (10 + 3i) (0 + 6i) = 0 + 0 + 60i -18 = -80 + 60i
8. (1 + i)4 - (1 - i)4 = ((1 + i)2)2 ((1 - i)2)2 = (1 + 2i -1)2 (1 2i -1)2
= (2i)2 (-2i)2 = -4 (-4)=0 = 0 + 0i
9. 12+i
i =
12+i
i .
i
i =
12i i2
i2 =
12i+1
1 = 1 12i
10. 3+4i
34i =
3+4i34i
. 3+4i3+4i
=9 +12i+12i16
9+16 =
7+24i
25 =
7
25 +
24
25i
-
) ( [ 3 1 ]
80087450770 / 01
11. i
2+3i =
0+i
2+3i.23i
23i =
0+2i0+3
4+9=
3+2i
13 =
3
13+
2
13i
12. (3+i
1+i)3
= (3+i
1+i .
1i
1i)3
= (3+i3i+1
1+1)3
= (42i
2)3
=(2 i)3 =(2 i)2. (2 i)=(4 4i 1). (2 i)
=(3 4i). (2 i)= 6 8i - 3i 4= 2 11i
13. 2+3i
1i.1+4i
4+i =
2+3i+8i12
44i+i+1 =
10+11i
53i =
10+11i
53i.5+3i
5+3i
= 10+11i
53i.5+3i
5+3i =
50+55i30i33
25+9 =
83+25i
34 =
83
34+
25i
34
14. (1 + i)3 + (1 - i)3 = (1 + i)2.(1 + i) + (1 - i)2.(1 - i)
= (1 + 2i -1).(1 + i) + (1 - 2i -1).(1 - i) = (2i).(1 + i) + (-2i).(1 - i) = 2i 2 + [-2i - 2] = 2i 2 2i - 2 = -4 = -4 + 0i
: y , x (2a) y + 5i = (2x + i)(x + 2i)
/y + 5i = 2x2 + xi + 4xi - 2
y + 5i = 2x2 + 5xi - 2 y + 5i = (2x2 2) + 5xi :
5x = 5 x = 1 y = 2x2 2 = 2 2 = 0
b) 8i = (x + 2i)(y + 2i) + 1 : /
-1 + 8i = xy + 2yi + 2xi - 4 -1 + 8i = (xy 4) + (2y + 2x)i :
-1 = xy 4 xy = 3 8 = 2y + 2x 4 = y + x y = 4 x .
: y
xy = 3 x(4 x) = 3 4x x2 = 3 x2 4x + 3 = 0 (x - 3)(x - 1) = 0
x = 3 y = 4 3 = 1 x = 1 y = 4 1 = 3
c) (1i
1+i) + (x+yi) = (1+2i)2
/
(1i
1+i .
1i
1i) + (x+yi) = (1 + 4i - 4) (
1ii1
1+1) + (x+yi) = -3 + 4i
(2i
2) + (x+yi) = -3 + 4i -i + (x+yi) = -3 + 4i
(x+yi) = -3 + 4i + i (x+yi) = -3 + 5i
-
) ( [ 3 1 ]
80087450770 / 07
: x = -3 , y = 5
d) 2i
1+i x +
3i
2+i y =
1
i
/
(2i
1+i .
1i
1i) x + (
3i
2+i .
2i
2i) y =
1
i (
2i2i1
1+1) x + (
62i3i1
4+1) y =
i4
i
(13i
2) x + (
55i
5) y = i3 (
13i
2) x + (
55i
5) y = - i
(1
2
3i
2) x + (1 i ) y = 0 i
1
2x
3x
2i + y - yi = 0 - i
(1
2x + y) + (
3x
2 y) i = 0 - i
:
1
2x + y = 0
3x
2 y = -1
---------------------
-x + 0 = -1 x = 1 :y
1
2. 1 + y = 0 y =
1
2
: (7
a) 1
(2i)2
1
(2+i)2=
8
25 i
L.H.S = 1
44i1
1
4+4i1 =
1
34i
1
3+4i =
1
34i.3+4i
3+4i
1
3+4i.34i
34i
=3+4i
9+16
34i
9+16 =
3+4i(34i)
25 =
3+4i3+4i
25=
8
25i = R.H.S
b) (1i)2
1+i+
(1+i)2
1i= 2882 2
L.H.S = 12i1
1+i+
1+2i1
1i =
2i
1+i+
2i
1i =
2i
1+i.1i
1i+
2i
1i.1+i
1+i
=2i2
1+1+
2i2
1+1=
2i2
2+
2i2
2 = -1 i + i -1 = -2 = R.H.S
c) (1 i) (1 i2)(1 i3) = 4
L.H.S = (1 i) (1 i2)(1 i3) = (1 i) (1 (-1))(1 (-i)) = (1 i) (1 + 1)(1 + i) = 2(1 i) (1 + i) = 2(1 + 1) = 4 = R.H.S
i2 = -1 i3 = - i
-
) ( [ 3 1 ]
80087450770 / 04
b , a a+bi 27, 823, 48, 03 (4 .
a) 85 = (81 + 4) = 81 - 4i2 = (9-2i)(9+2i) OR 85 = (4 + 81) = 4 - 81i2 = (2-9i)(2+9i) OR 85 = (49 + 36) = 49 - 36i2 = (7-6i)(7+6i) b) 41 = (25 + 16) = 25 - 16i2 = (5-4i)(5+4i)
c) 125 = (121 + 4) = 121 - 4i2 = (11-2i)(11+2i) OR 125 = (100 + 25) = 100 - 25i2 = (10-5i)(10+5i)
d) 29 = (25 + 4) = 25 - 4i2 = (5-2i)(5+2i)
y , x (36
x+yi ,
3+i
2i .
= c /3+i
2i
c = 3+i
2i c =
3i
2+i (c1
c2)
= (
c1
c2 )
6
x+yi =
3i
2+i
x + yi =6(2+i)
(3i)=
12 + 6i
3i .
3+i
3+i =
36 + 12i+18i6
9+1 =
30 + 30i
10 = 3+ 3i
x = 3 & y = 3
: 4ni-1/ 8
i4n-1 = i4n .i-1 = i-1 = i-1 .i4 = i3 = 0 i
/ 23i
2+i
3i
2i= 2
/
L.H.S = 3i
2+i
3i
2i =
3i
2+i.2i
2i
3i
2i.2+i
2+i =
32 i+3
2+1
32 i3
2+1
= 32 i+3
3
32 i3
3 = (2i+1) (2i1) = 2i + 1 2i + 1 = 2 = R.H.S
OR:
L.H.S = 3i
2+i
3i
2i =
3i(2i)3i(2+i)
(2+i) (2i) =
32 i+3 32 i+3
2+1=
6
2+1 = 2
=R.H.S
/75
x+yi
2+i
3i (8 2882). x , y ,
(2 5i) -+ (1 5(1 + i) .(2882 /4
-
) ( [ 4 1 ]
80087450770 / 05
a a : 1] [4
.a a a 2x = 8 a = 0 x = = 25 2x5 :
. x+yi c = a+bi .17- 25- /11
/
a) c2 = -25 c = 25 = 5i
b) c2 = -17 c = 17 = 17 i :
.a+bi -8 . x+yi -2 : -7
a. = 2 y - 2x . b. = 2 xy .
. x , y R -4 : /12
1) -3 + 4i c = -3 + 4i
x + yi c
x + yi = 3 + 4i (x + yi)2 = -3 + 4i
x2 + 2xyi y2 = -3 + 4i )x2 y2( + (2xy) i = -3 + 4i :
2xy = 4 y = 42x
y = 2x ..
x2 y2 = -3 .. :
x2 (2
x)2
= -3 x2 4
x2 = -3
x4 4 = -3x2 x2
x4 + 3x2 4 = 0 (x2 - 1)(x2 + 4) = 0 x2 + 4 = 0 x
x2 - 1 = 0 x2 = 1 x = 1 : x
x = 1 y = 2 c1 = 1 + 2i x = 1 y = 2 c2 = -1 - 2i
(2i+1) :
-
) ( [ 4 1 ]
80087450770 / 06
2) 8 + 6i c = 0 + 1i
x + yi c
x + yi = 8 + 6i (x + yi)2 = 0 + 1i
x2 + 2xyi y2 = 0 + 1i )x2 y2( + (2xy) i = 0 + 1i
2xy = 1 y = 62x
y = 3x ..
x2 y2 = 0 .. :
x2 (3
x)2
= 0 x2 9
x2 = 0
x4 7 = 8x2 x2
x4 - 8x2 9 = 0 (x2 - 9)(x2 + 1) = 0 x2 + 1 = 0 x
x2 - 9 = 0 x2 = 9 x = 3 : x
x = 3 y = 1 c1 = 3 + i x = 3 y = 1 c2 = -3 - i
(i+3) : 3) i c = 0 - i
x + yi c
x + yi = 0 i (x + yi)2 = 0 - i
x2 + 2xyi y2 = 0 I )x2 y2( + (2xy) i = 0 - i
2xy = -1 y = 12x
.. x2 y2 = 0 ..
:
x2 (1
2x)2
= 0 x2 1
4x2 = 0 x4
1
4 = 0 x2
x4 = 1
4 x =
1
2
: x
x =1
2 y =
1
2 1
2
= 1
2 c1 =
1
2 -
1
2 i
x = 1
2 y =
1
2 1
2
= 1
2 c2 =
1
2 +
1
2 i
) :
-
i)
-
) ( [ 4 1 ]
80087450770 / 00
4) 8i c = 0 + 8i
x + yi c
x + yi = 0 + 8i (x + yi)2 = 0 + 8i
x2 + 2xyi y2 = 0 + 8i )x2 y2( + (2xy) i = 0 + 8i :
2xy = 8 y = 4x ..
x2 y2 = 0 .. :
x2 (4
x)2
= 0 x2 16
x2 = 0 x4 16 = 0 x2
x4 16 = 0 x4 = 16 x = 2 : x
x = 2 y = 2 c1 = 2 + 2i x = 2 y = 2 c2 = -2 - 2i
(2i + 2) :
-
) ( [ 5 1 ]
80087450770 / 00
a 0 bx + c = 0 2ax + : 1 ] [ 5
a,b,c R , x =bb24ac
2a ,
4ac -2 b . , . 2x + 2 = 0 2x + /13
/x2 + 2x + 2 = 0 a = 1 , b = 2 , c = 2
x =bb24ac
2a
x =248
2 =
24
2=
2 2i
2 = -1 i
. (i -1-) (i +1-)
. 4x + 2x + 5 = 0 /14 /
x2 + 4x + 5 = 0 a = 1 , b = 4 , c = 5
x =bb24ac
2a
x =41620
2 =
44
2=
4 2i
2 = -2 i
. (i -2-) (i +2-) bx + c = 0 2ax + /
:a,b,c R a 0 . x-yi x+yi -8 : a 0 a -2
x2 + b
ax +
c
a = 0
:x2 ( ) x + (0 = (
: ,
= ,
=
(.-2i-2( )2i+2 : ) /15 /
(2 + 2i) +(-2 2i) = 0 (2 + 2i)(-2 2i) = - 4 - 4i 4i + 4 = -8i x2 ( )x + ( 0 = (
x2 0 x + (-8i) = 0 x2 - 8i = 0 x2 = 8i (. 4i-3 ) /16
/
(4i+3( )4i-3 ) (3 + 4i) +(3 4i) = 6 (3 + 4i)(3 4i) = 9 +12i 12i +16 = 25 x2 6x + 25 = 0
i = 2i 4 = 4/
-
) ( [ 5 1 ]
80087450770 / 07
i = 0-5x +7 2x /17 /
x2 5x +7-i = 0 a = 1 , b = -5 , c = 7-i
x =bb24ac
2a
x =5254(7i)
2 =
52528+4i
2 =
53+4i
2
/ = ,
4-1] ] . 9
12 :
c = -3 + 4i x + yi c
x + yi = 3 + 4i (x + yi)2 = -3 + 4i
x2 + 2xyi y2 = -3 + 4i )x2 y2( + (2xy) i = -3 + 4i :
2xy = 4 y = 42x
y = 2x ..
x2 y2 = -3 .. :
x2 (2
x)2
= -3 x2 4
x2 = -3
x4 4 = -3x2 x2
x4 + 3x2 4 = 0 (x2 - 1)(x2 + 4) = 0 x2 + 4 = 0 x
x2 - 1 = 0 x2 = 1 x = 1 : x
x = 1 y = 2 c1 = 1 + 2i x = 1 y = 2 c2 = -1 - 2i
(2i+1) : :
x =53+4i
2 x =
5(1+2i)
2
x =5+(1+2i)
2=
6+2i
2 = 3 + i
x =5(1+2i)
2=
42i
2 = 2 - i
{ i , 2 - i + 3}
-
) ( [ 5 1 ]
80087450770 / 18
1-2 (1
a) z2 = -12 z = 12 = 12 i = 23 i
S = {0+ i , 0- i} :
z2 = -12 z2 + 12 = 0 z2 12 i2 = 0 (z - 23 i) (z + 23 i) = 0
z - 23 i = 0 z = 23 i
z + 23 i = 0 z = 23 i
S = {0 i} ) (:
z2 = -12 z2 + 12 = 0 a = 1 , b = 0 , c = 12
z =bb24ac
2a
z =004(12)
2 =
48
2=
43 i
2 = 23 i
S = {0 i}
b) z2 3z + 3+ i = 0 z2 3z + 3+ i = 0 A=1 , B=-3 , C=3+i
z =bb24ac
2a
z =394(3+i)
2 =
39124i
2 =
334i
2
3 4i: (x + yi)2 = 3 4i
2xy = -4 y = 2x
x2 y2 = -3 x2 4
x2 = -3 x4 4 = -3x2 x2
x4 + 3x2 - 4 = 0 (x2 + 4)( x2 1) = 0 x2 1 = 0 x = 1
3 4i = (1 - 2i) :
z =3 34i
2 =
3 (1 2i)
2
z =3 + 1 2i
2 =
4 2i
2 = 2 - i
or z =3 1+ 2i
2 =
2+ 2i
2 = 1 + i
{i , 1+ i -2}
-
) ( [ 5 1 ]
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c) 2z2 5z + 13 = 0 2z2 5z + 13 = 0 a=2 , b=-5 , c=13
=
=( .)
. =
=
=
S = {
+
,
}
d) z2 + 2z + i(2-i) = 0 z2 + 2z + (2i - i2) = 0
z2 + 2z + (2i + 1) = 0 a = 1 , b = 2 , c =1+ 2i
z =bb24ac
2a
z =244(1+2i)
2 =
2448i
2=
208i
2
0 8i:
x + yi = 0 8i (x + yi)2 = 0 - 8i x2 + 2xyi y2 = 0 - 8i )x2 y2( + (2xy) i = 0 - 8i
:
2xy = -8 y = 4x
.. x2 y2 = 0 ..
:
x2 (4
x)2
= 0 x2 16
x2 = 0 x4 16 = 0 x2
x4 16 = 0 x4 = 16 x = 2 : x
x = 2 y = 2 x = 2 y = 2
:
z =208i
2 =
2(22i)
2=
2 + 2 2i
2 = - i
or z =2 2 + 2i
2 =
4 + 2i
2= -2 + i
S = {- i , -2+ i}
) (:
z2 + 2z + i(2-i) = 0 z2 + 2z + 2i - i2 = 0 (z2 i2) + (2z + 2i) = 0 (z2 i2) + 2(z + i) = 0 (z - i)(z + i) + 2(z + i) = 0 (z + i) [(z i) + 2] = 0 z + i = 0 z = -i or (z i) + 2 = 0 z i = -2 z = -2 + i S = {- i , -2+ i}
0 8i = (2 - 2i)
-
) ( [ 5 1 ]
80087450770 / 11
e) 4z2 + 25 = 0 4z2 25i2 = 0 z2 = 25
4i2 z =
5
2i
S = { 0
}
:4z2 + 25 = 0 a = 4 , b = 8 , c = 23
z =bb24ac
2a
z =004(4 .25)
2 .4 =
400
8=
400 i
8 =
20 i
8 =
5 i
2
S = { 0
}
f) z2 - 2zi + 3 = 0 : /
z2 - 2zi + 3 = 0 a=1 , b=-2i , c=3
z =bb24ac
2a
z =2i 4i24(3)
2 =
2i412
2 =
2i 16
2 =
2i 16 i
2=
2i 4 i
2=
2i 4 i
2
z =2i+ 4 i
2 = 3i
or z = 2i 4 i2
= -i S = {0 - i , 0 + 3i}
(:2i) /
z2 - 2zi + 3 = 0 z2 - 2zi 3i2 = 0 (z - 3i)(z + i) = 0 z - 3i = 0 z = 3i or z + i = 0 z = -i S = {0 - i , 0 + 3i}
/ :
z2 - 2zi + 3 = 0 z2 - 2zi = -3 z2 - 2zi + i2 = -3 + i2 (z - i) (z - i) = -3 -1 = -4 (z - i)2 = -4 z - i = 2i z = i 2i Neither: z = i + 2i z = 3i OR : z = i 2i z = -i S = {0 - i , 0 + 3i}
: m.L (2a) m = 1 +2i , L = 1- i m+L = (1 + 2i) + (1 - i) = 2 + i m.L = (1 + 2i)(1 - i) = 1 + 2i i + 2 m.L = 3 + i x2 (2 + i) x + (3 + i) = 0
-
) ( [ 5 1 ]
80087450770 / 17
b) m = 3 i
1+ i , L = (3- 2i)2
m = 3 i
1+ i =
3 i
1+ i .
1 i
1 i =
(3 i)(1 i)
1+ 1 =
3i3i1
2 =
24i
2 = 1 2i
L = (3- 2i)2 = 9 12i 4 = 5 12i m+L = (1 - 2i) + (5 - 12i) = 6 - 14i m.L = (1-2i)(5-12i) = 5 10i 12i 24 = -19 - 22i
:x2 (6 - 14i) x + (-19 - 22i) = 0
: (3a) -6i
c = 0 - 6i x + yi c
x + yi = 0 6i (x + yi)2 = 0 - 6i
x2 + 2xyi y2 = -0 - 6i )x2 y2( + (2xy) i = 0 - 6i :
2xy = -6 y = 62x
y = 3x
... x2 y2 = 0 ..
:
x2 (3
x)2
= 0 x2 9
x2 = 0
x4 9 = 0 x2
x4 = 9 x2 = 3 x = 3 : x
x = 3 y =3
3= 3 c1 = 3 3 i
x = 3 y =3
3= 3 c2 = -3 + 3 i
(i - ) : b) 7+24i c = 7 + 24i
x + yi c
x + yi = 7 + 24i (x + yi)2 = 7 + 24i
x2 + 2xyi y2 = 7 + 24i )x2 y2( + (2xy) i = 7 + 24i :
2xy = 24 y = 242x
y = 12x
... x2 y2 = 7 ..
:
x2 (12
x)2
= 7 x2 144
x2 = 7 x4 144 = 7x2 x2
-
) ( [ 5 1 ]
80087450770 / 14
x4 7x2 - 144 = 0 (x2 - 16)( x2 + 9) = 0 x2 + 9 = 0
x2 16 = 0 x2 = 16 x = 4 : x
x = 4 y = 3 c1 = 4 + 3i x = 4 y = 3 c2 = -4 - 3i
(3i + ) :
c) 4
13 i
4
13 i =
4
13 i .
1+3 i
1+3 i =
4(1+3 i)
1+3=
4(1+3 i)
4 = 1 + 3 i
c = 1 + 3 i x + yi c
x + yi = 1 + 3 i (x + yi)2 = 1 + 3 i
x2 + 2xyi y2 = 1 + 3 i )x2 y2( + (2xy) i = 1 + 3 i :
2xy = 3 y = 3
2x ....
x2 y2 = 1 ..
:
x2 (3
2x)2
= 1 x2 3
4x2 = 1 4x4 3 = 4x2 4x20
4x4 4x2 - 3 = 0 (2x2 - 3)(2x2 + 1) = 0 2x2 + 1 = 0
2x2 3 = 0 x2 = 3
2 x =
3
2
: x
x =3
2 y =
3
2(3
2) =
32
23=
2
2 =
1
2
x = 3
2 y =
1
2
) :
+
i)
22+1- / /
c = -1+22 = -1+22 i x + yi c
x + yi = 1 + 22 i (x + yi)2 = -1+22 i
-
) ( [ 5 1 ]
80087450770 / 15
x2 + 2xyi y2 = -1+22 i )x2 y2( + (2xy) i = -1+22 i :
2xy =22 y =22
2x y =
2
x...
x2 y2 = -1 .. :
x2 (2
x)2
= -1 x2 2
x2 = -1 x4 2 = -x2 x2
x4 + x2 2 = 0 (x2 - 1)( x2 + 2) = 0 x2 + 2 = 0 x R x2 1 = 0 x2 = 1 x = 1
: x
x = 1 y = 2
x = 1 y = 2
(i + ) :
: (4a) i
i , -i i + (-i) = 0 i. (-i) = 1 x2 - (0) x + 1 = 0 x2 + 1 = 0
b) 5 i i , 5-i+5
5+i + (5-i) = 10 (5+i)(5-i)=25 + 1= 26 x2 - 10x + 26 = 0
c) 2+ 3i
4
2+ 3i
4 ,
2 3i
4
2 3i
4 +
2+ 3i
4 =
22
4 =
2
2=
1
2
2 3i
4 .
2+ 3i
4 =
2+9
16 =
11
16
x2 - 1
2 x +
11
16 0 =
-
) ( [ 5 1 ]
80087450770 / 16
ax + (5+5i) = 0 2x : i+3 (5 a 2x = , i 1x + 3 =
=
x2
x1 . x2 = 5+5i
1 = 5+5i (3 + i) . x2 = 5 + 5i
x2 = 5+5i
3+i =
5+5i
3+i .
3i
3i =
15+15i5i+5
9+1 =
20+10i
10 = 2 + i
= = x
x2
x1 + x2 = (a)
1 (2 + i) + (3 + i) = a a = 5 + 2i
:
a
(3 + i)2 a(3 + i) + (5 + 5i) = 0 (9 + 6i - 1) a(3 + i) + (5 + 5i) = 0 (8 + 6i) a(3 + i) + (5 + 5i) = 0 a(3 + i) = (8 + 6i) + (5 + 5i) = 13 + 11i
a = 13+11i
3+i =
13+11i
3+i .
3i
3i =
39+33i13i+11
9+1 =
50+20i
10 = 5 + 2i
: : 2x =
(3 + i) + x2 = 5 + 2i x2 = (5 + 2i) (3 + i) x2 = 2 + i
-
) ( [ 6 1 ]
80087450770 / 10
a : 1 ] [ 6
a a 3x = 3
, - )( :
3z 1= 3z = : -8 3z = 0 1 - 3z -= 8 : -2 ) ( : -7
z3 1 = 0 (z 1)(z2 + z +1) = 0 z 1 = 0 z = 1
z2 + z +1 = 0 z = 114
2 =
13
2 =
13 i
2
: -4
1 , 1
2+
3
2i ,
1
2
3
2i
, . 8 -8 : -2
1 + )1
2+
3
2i( + )
1
2
3
2i( = 0
8 = -7
)1
2+
3
2i()
1
2
3
2i( = 8
= -4
(1
2+
3
2i)2 =
1
2
3
2i
(1
2
3
2i)2 =
1
2+
3
2i
) 1
2
3
2i) ,(
1
2+
3
2i) w ( Omega)
w , w , 21 2w :
1- w3 = 1 2- 1 + w + w2 = 0
: 2
1 + w = -w2 1 + w2 = -w w + w2 = -1 w = -1 - w2 w2 = -1 - w 1 = -w2 - w
2w , w . : a , a :
a3
, a3
w , a3
w2 -:
2w , 2w , 22 : 0 - , 2w-w , -1 : -1
-
) ( [ 6 1 ]
80087450770 / 10
:w w3 = 1 , w4 = w3 . w = w w5 = w3 . w2 = w2 w6 = w3 . w3 = 1
w , w , 21 (w) -: , 7
w20 = w18 . w2 = (w3)6 . w2 = w2 w100 = w99 . w = (w3)33 . w = w w3n = (w3)n = 1 n
w3n-1 = (w3)n .w-1= w-1 = 1
w =
w3
w = w2
w-4 = 1
w4 =
1
w3 .w =
1
w = w2
w w/ .w 3
or w-4 = w6 . w-4 = w2 w-5 = w6 . w-5 = w w-6 = w6 . w-6 = w0 = 1 w-20 = w21 . w-20 = w w-31 = w33 . w-31 = w2
r = 0 , 1 , 2 , r= w 3n+rw n : :
w33 = w3(11) + 0 = w0 = 1 w25 = w3(8) + 1 = w1 = w w-58 = w3(-20) + 2 = w2
w 3 w : : /18
a) (3 + 2w + 2w2)20 = [3 + 2(w + w2)]20 2 = [3 + 2(-1)]20 w + w2 = -1 = [3 2]20 = 1
. /
b) (1 - 3w - 3w2)4 = [1 3(w + w2)]4 7 = [1 3(w + w2)]4 w + w2 = -1 = [1 3(-1)]4 = [1 + 3]4 = 44 = 256
c) (3 + 4w + 5w2)2 w -1 -= 2 w /
= [3 + 4w + 5(-1 w)]2 = [3 + 4w - 5 5w]2 = [-2 - w]2
= 4 + 4w + w2 4 = 4(1 + w) + w2 1 + w = -w2 = 4(-w2) + w2 = -4w2 + w2 = -3w2
-
) ( [ 6 1 ]
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. 2w w /
: /191) w7 + w5 + 1 = 0 L.H.S= w7 + w5 +1= w6. w + w3 . w2 +1 = w + w2 + 1 = 0 = R.H.S
2) (5+3w+3w2)2 = -4(2+w+2w2)3 = 4 L.H.S = (5 + 3w + 3w2)2
= [5 + 3(w + w2)]2 7 = [5 + 3(-1)]2 w + w2 = -1 = [5 - 3]2 = 22 = 4 = R.H.S
M.H.S = -4(2+w+2w2)3 = -4(w + 2 + 2w2)3 2 = -4[(w + 2(1+ w2)]3 1+w2 = -w = -4[(w + 2(-w)]3 = -4[w 2w]3 = -4[-w]3 = -4[-1] = 4 = L.H.S = R.H.S
: /211) 1 iw , 1 - iw2 2882 (1 iw) + (1 iw2) = 2 iw iw2 = 2 i(w + w2) = 2 i(-1) = 2 + i (1 iw) (1 iw2) =8 iw iw2 + i2w3 = 8 iw iw2 -1 = iw iw2 = i(w + w2) = i(1) = i x2 (2+i) x + i = 0
2) 3w + w2 , w + 3w2 (3w + w2) + (w + 3w2) = 4w + 4w2 = 4(w + w2)= 4(-1) = -4
(3w + w2)(w + 3w2) = 3w2 + w3 + 9w3 + 3w4 = 3w2 + 1 + 9 + 3w3.w = 3w2 + 10+ 3w =10 + 3w + 3w2 = 10 + 3(w + w2) = 10 + 3(-1) = 10 3 = 7 x2 + 4x + 7= 0
3) 1-2w , 1-2w2 (1-2w) + (1-2w2) = 2 - 2w - 2w2 = 2 - 2(w + w2) = 2 - 2(-1) = 4 (1-2w)(1-2w2) = 1 2w -2w2 +4w3 = 1 2w -2w2 + 4 = 5 2w 2w2 = 5 2(w + w2) = 5 2(-1) = 7
x2 - 4x + 7= 0 3 2/
.
-
) ( [ 6 1 ]
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4) 2iw 3w2
i , 3iw
2w2
i
: /
2iw 3w2
0+i.
0i
0i = 2iw + 3iw2
3iw 2w2
0+i.
0i
0i = 3iw + 2iw2
(2iw + 3iw2) + (3iw + 2iw2) = 5iw + 5iw2 = 5i(w + w2)= 5i(-1) = 5i (2iw+3iw2)(3iw+2iw2) = 6w2 4 9 6w = 13 6(w + w2)= 13 + 6 = 7
x2 + 5ix 7 = 0
5) 2
1w ,
2
1w2
(2
1w) + (
2
1w2) =
2(1w2)+ 2(1w)
(1w)(1w2) =
22w2+ 22w
(1w)(1w2) =
42w22w
1ww2+1 =
42(w2+w)
2(w+w2)
= 4+2
2+1=
6
3 2 =
(2
1w)(
2
1w2) =
4
1w w2+ 1=
4
2(w+ w2)=
4
2+1 =
4
3
x2 2x + 4
3 0 =
/21 a + bw + cw2
b + cw + aw2
) , ( / / . 3w w
a + bw + cw2
b + cw + aw2 =
aw3+ bw + cw2
b + cw + aw2 =
w(aw2+ b + cw)
b + cw + aw2 = w
(2 /221
wa+bi=(1+2w+ , R a,b
b 2a +2 1 = : (8 .a+bi (2
/
1) a + bi = (1 + 2w + 1
w)2 = (1 + 2w +
w3
w)2 = (1 + 2w + w2)2 = ( w + 2w)2 = w2
a + bi = w2
a2 + b2 = a2 b2 i2 = (a - bi)(a + bi) : 2w w
a2 + b2 = w . w2 = w7 = 1
( 2w , w 1
2
3
2 i(
:w 2w , (2w + w2 = -1 w . w2 = 1 x2 - x + 1 = 0
-
) ( [ 6 1 ]
80087450770 / 70
1 - 3 : (1
a) w64 = w63 . w = (w3)21. w = w b) w325 = w327. w325 = w2
c) 1
(1+w32)12 =
1
(1+w33.w32)12 =
1
(1+w)12 =
1
(w2)12 =
1
w24 = 1
d) (1+w2)4 = (-w)4 = 1
(w)4 =
1
w4 =
w6
w4 = w2
e) w9n+5 , n N w9n+5 = w9n. w5 = (w3)3n. w5 = w5 = w3 .w2 = w2
: (2a) 1+w2 , 1+w
/(1+w2) + (1+w) = 2 + w + w2 = 2 8 = 1
(1+w2)(1+w) = 1 + w2 + w + w3 = 1 + w + w2 + 1 = 2 + -1 = 1
x2 - x + 1 = 0
b) w
2w2 ,
w2
2w
(w
2w2) + (
w2
2w (
= w(2w) + w2(2w2)
(2w2)(2w) =
2ww2+ 2w2w4
42w22w+w3=
2w+ w2w
52w22w=
w+ w2
52(w2+w) =
1
5+2 =
1
7
(w
2w2)(
w2
2w)
= w3
42w22w+w3=
1
52w22w=
1
52(w2+w) =
1
5+2 =
1
7
x2 + 1
7 x +
1
7 = 0 7x2 + x + 1 = 0
c) 3i
w2 ,
3w2
i
: /3i
w2 .
w
w = 3iw
3w2
i .
i
i = 3iw2
(3iw + 3iw2) = 3i(w + w2) = -3i
3iw . 3iw2 = 9i2w3 = 9 x2 + 3ix 9 = 0
-
) ( [ 6 1 ]
80087450770 / 71
: z + 1 = 02 z + : (3
1+3z10+3z11
13z73z8
z /z2 + z + 1 = 0 a = 1 , b = 1 , c = 1
z =114
2 =
13
2=
13 i
2 z =
1
2
3
2 i z = w or w2
z = w
1+3z10+3z11
13z73z8=
1+3w10+3w11
13w73w8 =
1+3w+3w2
13w3w2 =
1+3(w+w2)
13(w+w2)
= 13
1+3 =
2
4 =
1
2
2z = w
1+3z10+3z11
13z73z8 =
1+3(w2)10+3(w2)11
13(w2)73(w2)8 =
1+3w20+3w22
13w143w16=
1+3w2+3w
13w23w
= 1+3(w2+w)
13(w2+w) =
13
1+3=
2
4 =
1
2
: (4
a) (1
2+w
1
2+w2)2 =
1
3
L.H.S = (1
2+w
1
2+w2)2=(
(2+w2) (2+w)
(2+w)(2+w2))2
= (w2 w
4+2w+2w2+w3)2
=(w2 w
5+2(w+w2))2
=(w2 w
52)2
=(w2 w
3)2
= w4 2w3+w2
3 =
w+w2 2
3=
3
9 =
1
3 = R.H.S
b) w14+w71
w10+w52 =
2
3
L.H.S = w14+w71
w10+w52 =
w2+w1
w+w22=
11
12 =
2
3 =
2
3 = R.H.S
c) (1 2
w2+ w2) (1 + w
5
w8 2884 18 = (
L.H.S = (1 2
w2+ w2) (1 + w
5
w) = (1
2w3
w2+ w2) (1 + w
5w3
w)
= (1 2w + w2)(1 + w 5w2) = (w 2w)(w2 5w2) = (3w)(6w2) = 18 w3 = 18 = R.H.S
d) (1+ w2)3 + (1+ w)3 = -2 L.H.S = (1+ w2)3 + (1+ w)3 = (w)3 + (w2)3 = w3 w6
= w3 (w3)2 = 1 1 = 2 = R.H.S
= 1 3w 2w1+w =
w= 21+w
-
) ( [ 7 1 ]
80087450770 / 77
z : 1 ] [ 7(x,y) z(x,y) ) ( z ,
= {(x, y) x , y R} .
(x,y) , 2or E 2(R(
.
z = x+yi (P(x,y (O(0,0 .Y Axis y X-Axis x
: /231) (3 + 4i) + (5 + 2i)
(3 + 4i) + (5 + 2i) = 8 + 6i / ,
2P 1P 3P
3,P2,P1O,P O . 1P(3,4) 4i+3 2P(5,2) 2i+5
, . 3P
2) (6 - 2i) - (2 - 5i( (6 - 2i) - (2 - 5i) = (6 - 2i) + (-2 + 5i)= 4 + 3i
6,1P)-(2 2i-6 2P)-5,2( -i5+2
y
x
O(0,0)
P(x,y)
y
x
O(0,0) 2)-(6,1P
2,5)-(2P
(4,3)3P
y
x O(0,0)
(5,2)2P
(3,4)1P (8,6)3P
-
) ( [ 7 1 ]
80087450770 / 74
1 4
: (1z1 = 2+3i , z2 = -1+3i , z3 = 1-i , z4 = i
z2 = -1+3i = (-1,3) -z2 = 1-3i = (1,-3)
z1 = 2+3i = (2,3) -z1 = -2-3i = (-2,-3)
z4 = i =0 + i = (0, 1) -z4 = - i = 0 - i = (0,-1)
z3 = 1- i = (1,-1) -z3 = -1+ i = (-1,1)
: (2z1 =3+3i , z2 =-7+2i , z3 =1-i , z4 = -2i
z2 = -3 + 2i = (-3, 2) z2 = -3 - 2i = (-3,-2)
z1 = 5 + 3i = (5, 3) z1 = 5 - 3i = (5,-3)
z4 = -2i = 0 - 2i = (0, -2) z4 = 2i = 0 + 2i = (0,2)
z3 = 1 - i = (1, -1) z3 = 1 + i = (1,1)
(-1,1)
(1,-1)
(5,-3)
(5, 3)
(-3,-2)
(-3, 2)
(1,-1)
(1, 1)
(0,-2)
(0, 2)
(-1,3)
(1,-3)
(0,-1)
(0, 1)
1z
1z-
(2,3)
(-2,-3)
-
) ( [ 7 1 ]
80087450770 / 75
z , z , z : z = 4+2i (3 /
z = 4 + 2i = (4 , 2) z = 4 2i = (4 , -2) z = -4 2i = (-4 , -2)
: 2i - = 4 1z ,2i + 1 = 2z (4-3z2 , 2z1 , z1 z2 , z1 + z2
z1 = 4 - 2i 2z1 = 2(4 - 2i) = 8 - 4i = (8,-4)
z2 = 1 + 2i -3z2 = -3(1 + 2i) = -3 -6i = (-3,-6)
z1 = 4 - 2i , z2 = 1 + 2i
z1 + z2 = (4 - 2i) + (1 + 2i) = 5 - 0i = (5, 0)
z1 = 4 - 2i , z2 = 1 + 2i z1 - z2 = (4 - 2i) (1 + 2i) = (4 - 2i) + (-1 - 2i) = 3 - 4i = (3,-4)
(4, -2)
(4, 2)
(-4, -2)
(-3,-6)
(3,-4)
(-1,-2) (4,-2)
(4,-2)
(5, 0)
(1, 2)
(8,-4)
-
) ( [ 8 1 ]
80087450770 / 76
(P(x,y z = x +yi : 1 ] [ 8
(r,) P O ) ) OX .
z r (Mod z) z :
r = z = x2 + y2
OP
:
cos = x
r =
x
z R(z) = x = r.cos
sin = y
r =
y
z (z) = y = r.sin
(z) (R(z :I(z) (z)
= arg (z)
[0, 2) , + 2n )n ) .
- : ) ( . -8
. -2
, 0} -7
2 , ,
3
2 {
.
: /24
1) z1 = 1- 3i
z1 = 1- 3i =(1,- 3)
Mod(z) = z = r = x2 + y2 = 1 + 3 = 2 unit
cos = x
z =
1
2 , sin =
y
z =
3
2
=
- :
= arg(z) = 2 -
3 =
5
3
0
sin
0 1 0 -1
cos
1 0 -1 0
Y
X O
P(x,y)
r y
x
+, y +x sin+ , cos+ =
+, y x sin+ , cos
=
, y +x sin , cos+
= 2
, y x sin , cos
= +
-
) ( [ 8 1 ]
80087450770 / 70
2) -1-i z2 = -1- i =(-1, -1)
r = Mod(z) = z = x2 + y2 = 1 + 1 = 2 unit
cos = x
z =
1
2 , sin =
y
z =
1
2
=
,
= arg(z) = +
4 =
5
4
3) i z3 = 0 + i =(0, 1)
r = Mod(z) = z = x2 + y2 = 0 + 1 = 1 unit
cos = x
z = 0 , sin =
y
z =
1
1 = 1
=
2
2 z /25
6 .z,
/
r = z = 2 , = arg(z) =
6
= cos : cos x x
r
x = r . cos = 2 (cos
6) = 2 (
3
2) = 3
= sin : sin y y
r
y = r . sin = 2 (sin
6) = 2 (
1
2) = 1
z = x + yi = 3 + i
/26
4
1
2.
= r :- sin sin y
r
r = y
sin . =
1
2
sin
4
= 1
2
1
2
= 1
= x :- cos cos x
r
x = r.cos = 1 . cos
4 = 1(
1
2) =
1
2
z = 1
2 +
1
2 i
-
) ( [ 8 1 ]
80087450770 / 70
: /271) -2+2i = (-2,2)
r = Mod(z) = z = x2 + y2 = 4 + 4 = 8 = 2 2
cos = x
r =
2
22 =
1
2 , sin =
y
r =
2
22 =
1
2
=
,
= arg(z) = -
4 =
3
4
- :
z = r (cos + i sin ) = 22 (cos 3
4 + i sin
3
4)
2) 23 - 2i = (23 , -2) 2882
r = Mod(z) = z = x2 + y2 = 12 + 4 = 16 = 4
cos = x
r =
23
4 =
3
2 , sin =
y
r =
2
4 =
1
2
=
,
= arg(z) = 2 -
6 =
11
6
z = r (cos + i sin ) = 4 (cos 11
6 + i sin
11
6 (
. z = 0 (8 z = x+yi (2
: Polar form
x = r cos , y = r sin
z = r cos + i r sin
= r(cos + i sin )
z = z[cos (arg z)+ i sin (arg z)] z (r = Mod(z) = z , = arg(z:
-
) ( [ 8 1 ]
80087450770 / 77
: /28
b) i
a) 1
d) -i
c) -1
:3 = 3 . 1 = 3(cos 0 + i sin 0)
-2 = 2 . (-1) = 2(cos + i sin )
5i = 5 . i = 5(cos
2 + i sin
2)
-7i = 7 .(-i) = 7(cos 3
2 + i sin
3
2)
- : 1 = (cos 0 + i sin 0)
-1 = (cos + i sin )
i = (cos
2 + i sin
2)
-i = 1(cos 3
2 + i sin
3
2)
(1, 0)
Pz1 = (1,0) = 1+0i mod z1 = 1 arg z1 = 0
z1 = 1(cos 0 + i sin 0)
(0, 1)
Pz2 = (0,1) = 0+1i mod z2 = 0
arg z2 =
2
z2 = 1(cos
2 + i sin
2)
(-1, 0)
Pz3 = (-1,0) = -1+0i mod z3 = 1
arg z3 =
z3 = 1(cos + i sin ) (0, -1)
Pz4 = (0,-1) = 0- i mod z4 = 1
arg z4 = 3
2
z4 = 1(cos 3
2 + i sin
3
2)
-
) ( [ 9 1 ]
80087450770 / 48
: 1z ,2z : 1 ] [ 9z1 = cos + i sin z2 = cos + i sin
: 2z . 1z z1 . z2 = (cos + i sin)( cos + i sin)
= cos cos + i cos sin + i sin cos - sin sin z1 . z2 = (cos cos - sin sin) + i (cos sin + sin cos) z1 . z2 = cos( + ) + i sin( + )
= : z1 . z2 = cos(2) + i sin(2)
:
cos(2) + i sin(2) = (cos + i sin)2 :
R.H.S = (cos + i sin)2 = cos2 + 2i sin cos - sin2 =(cos2 - sin2) + i(2sin cos) = cos2 + i sin2 = L.H.S
:
(4 - : /293
8+ i sin
3
8(cos
(cos 3
8 + i sin
3
8)4 = cos 4(
3
8) + i sin4(
3
8) = cos
3
2 + i sin
3
2= 0 + i(1)
(cos 3
8 + i sin
3
8)4 = i
)ni sin - ncos= n)sini - cos : N n ,R / 78
L.H.S = (cos - i sin)n = [cos + i (-sin)]n = [cos() + i sin()]n
= :
= [cos + i sin ]n = cos n + i sin n = cos (n) + i sin (n)
= cos n - i sin n = R.H.S
(i + 1)11 /31
z = (1+ i) = (1 , 1) :z
mod(z) = r = 2
cos = 1
2 , sin =
1
2
: z
z = r )cos + i sin ( =
N n ,R )ni sin + ncos= n)sin+ i cos :
N n ,R : : + i sin z = r(cos(
zn = rn (cos + i sin)n = rn (cos n + i sin n)
-
) ( [ 9 1 ]
80087450770 / 40
z = 2 )cos
+ i sin
(
:
zn = rn (cos n + i sin n)
z11 = (2)11 (cos
+ i sin
)
z11 = (2)112 (cos
+ i sin
)
z11 = (2)512 (
1
2 +
1
2 i)
z11 = 32 2 (1
2 +
1
2 i) = 32 (-1+ i)
(1 + i)11 = 32 (-1+ i)
:
(cos + i sin)-n = cos(n) - i sin(n)
3x , x 0 = 1 + /32
x3 + 1 = 0 x3 = -1 : 20 1-
x = (cos + i sin )131
:
= , n = 3
x = (cos+2k
n + i sin
+2k
n) k = 0 , 1 , 2
k = 0 x = (cos
3+ i sin
3) =
1
2+
3
2i
k = 1 x = (cos + i sin )= 1 + i(0) = 1
k = 2 x = (cos5
3+ i sin
5
3 (
x = cos (2
3) + i sin (2
3) = cos(
3) i sin(
3) =
1
2
3
2i
, 1 } : 1
2+
3
2i ,
1
2
3
2i }
N , n > 1 n ,R : :
= (
+
+
+
)
k = 0 , 1 , 2 , , n-1:
=/
=
+
cos 3
4 = cos (
4) = -cos
4 =
sin 3
4 = sin (
4) = sin
4 =
)i sin - cos= ) -(i sin + )-(cos= 1-)sin+ i cos /
-
) ( [ 9 1 ]
80087450770 / 41
3) /33 + i)2
.
z = 3 / + i z = 3 + i = (3 , 1)
:z
mod(z) = r = 3 + 1 = 2
cos = 3
2 , sin =
1
2 , arg(z) =
6
z = 2 )cos
6 + i sin
6 z )
: 2z
z2 = 22 )cos
6 + i sin
6(2 = 4 )cos
3 + i sin
3(
: 2z
z252 = [4 (cos
3 + i sin
3)]
152 = 4
152 (cos
3+ i sin
3)
152
= 45
(cos
3 + i sin
3)
152
: =
3 , n = 5
k = 0 z252 = 4
5 (cos
15 + i sin
15)
k = 8 z252 = 4
5 (cos
7
15+ i sin
7
15)
k = 2 z252 = 4
5 (cos
13
15+ i sin
13
15)
k = 7 z252 = 4
5 (cos
19
15+ i sin
19
15)
k = 4 z252 = 4
5 (cos
25
15+ i sin
25
15) = 4
5 (cos
5
3+ i sin
5
3)
)(
2+ i sin 2(cos2= r 2z( 1+ i sin 1(cos1= r 1z( : :
z1 . z2 = r1. r2[cos(1+2)+ i sin(1+2)]
( /
6+ i sin
6(cos2 = 1z )
2
3+ i sin
2
3(cos7= 2z 2 . z1 z
. /
z1 . z2= 2(3)[cos(
6 +
2
3)+ i sin(
6 +
2
3)] = 6 [cos(
5
6)+ i sin(
5
6 [(
= 6 [3
2+ i (
1
2)] = 33 + 3i
3) : / + i)25
-
) ( [ 9 1 ]
80087450770 / 47
)(
2+ i sin 2(cos2= r 2z( 1+ i sin 1(cos1= r 1z( : :
z1
z2 =
r1
r2 [cos(1-2)+ i sin(1-2)]
( / 5
6+ i sin
5
6= 4(cos 1z ,)
6+ i sin
6(cos7 = 2z
z1
z2
. /
z1
z2 =
4
3 [cos(
5
6 -
6)+ i sin(
5
6 -
6)] =
4
3 [cos(
4
6)+ i sin(
4
6)]
= 4
3 [cos(
2
3)+ i sin(
2
3 [(
z1
z2 =
4
3 [-cos(
3)+ i sin(
3)] =
4
3 (
1
2 +
3
2 i) =
2
3 +
2
3 i
1 - 5 : -8
a) [cos5
24 + i sin
5
24]
4
= cos 4 (5
24) + i sin 4 (
5
24) = cos (
5
6) + i sin (
5
6)
= cos (
6) + i sin (
6) = cos (
6) + i sin (
6) =
3
2+ 1
2 i
b) [cos7
12 + sin
7
12]
3 = cos 3 (
7
12) i sin 3 (
7
12) = cos (
7
4) i sin (
7
4)
= cos (2
4) i sin (2
4) = cos (
4) + i sin (
4) =
1
2+
1
2 i
) ( : -2
a) (1 i)7 8 2887 , 2882
: (i-1) /z = 1 i = (1,-1)
r = Mod(z) = x2 + y2 = 1 + 1 = 2unit
cos = x
r =
1
2 , sin =
y
r=
1
2
=
,
= arg(z) = 2
4=
7
4
: z
z = 2 )cos
+ i sin
(
:
z7 = (2)7 (cos
+ i sin
)7 = (2)
7 (cos
+ i sin
)
-
) ( [ 9 1 ]
80087450770 / 44
z7 = 82 (cos
+ i sin
)
z7 = 82 (1
2 +
1
2 i)
(1 - i)7 = 8 + 8 i / " " :
(1 - i)7 = [(1 - i)2]3 (1- i) = (1 - 2i - 1)3 (1- i) = (-2i)3 (1- i) = -8 i3 (1- i) = 8i (1- i) = 8 + 8i
b) (3 + i) 2887 9-
/
z = 3 + i = (3, 1)
r = Mod(z) = x2 + y2 = 3 + 1 = 2unit
cos = x
r =
3
2 , sin =
y
r=
1
2
=
,
= arg(z) =
6
: z
z = 2 )cos
+ i sin
(
:
z-9 = (2)-9 (cos
+ i sin
)-9 = (
1
29) (cos
- i sin
) =
1
512 (cos
- i sin
)
z-9 = 1
512 (0 (-i)) =
1
512 i
: -7
a) (cos2 + i sin2)5
(cos3 + i sin3)3 =
[(cos + i sin)2]5
[(cos + i sin)3]3 =
(cos + i sin)10
(cos + i sin)9 = cos + i sin
b) (cos + i sin)8(cos - i sin)4 ..
: /
(cos + i sin)8(cos - i sin)4 = (cos + i sin)8(cos + i sin)-4
= (cos + i sin)4 = cos4 + i sin4 :
(cos + i sin)8(cos - i sin)4 = (cos 8 + i sin 8)(cos4 + i sin 4) =(cos 8 cos 4 + i cos 8 sin 4 + i cos 4 sin 8 + i2 sin 8 sin 4) =(cos 8 cos 4 sin 8 sin 4 + i (cos 8 sin 4 + cos 4 sin 8)) =(cos(8 4) + i ( sin 8 4)) = cos4 + i sin4
Hint: x4y4 = (x.y)4
/49
4=
49
4 12 =
2= 8 (2)6(2)= 7(2)
-
) ( [ 9 1 ]
80087450770 / 45
.8-4 i 3+1- -4 : /
z =1 + 3 i z = = 1 + 3 i = (-1 , 3)
:z
mod(z) = r = 1 + 3 = 2 unit
cos = 1
2 , sin =
3
2
=
,
arg(z) =
3 =
2
3
: z
z = 2 )cos 2
3 + i sin
2
3(
: z h
z = 2 (cos 2
3 + i sin
2
3) = 2(cos
2 3 +2k
2+ isin
2 3 +2k
2)
k = 0 , 1
k = 0 z = 2 (cos 2
6+ i sin
2
6) = 2 (cos
3+ i sin
3)
= 2 ( 1
2+
3
2i) =
1
2+
3
2i
k = 8 z = 2 (cos 8
6+ i sin
8
6) = 2 (cos
4
3+ i sin
4
3)
= 2 (cos
3 i sin
3) = 2 (
1
2
3
2i) =
1
2
3
2i
1 + 3 i = (1
2+
3
2i)
:8-4/
1 + 3 i = x+ y i
1 + 3 i = (x + y i)2 1 + 3 i = x2 + 2xy i y2
:
2xy = 3 y = 3
2x
x2 y2 = -1 . :
x2 (3
2x)2 = -1 x2
3
4x2 = -1
4x4 3 = -4x2 4x2 4x4 + 4x2 3 = 0 (2x2 - 1) (2x2 + 3) = 0 2x2 + 3 = 0 x
-
) ( [ 9 1 ]
80087450770 / 46
2x2 1 = 0
2x2 = 1 x2 = 1
2 x =
1
2
:y 8 x
x = 1
2 y =
3
2 . 1
2
= 3
2
x = 1
2 y =
3
2 . 1
2
= 3
2
1 + 3 i = (1
2+
3
2i)
.27i -3
27i = 27 (cos
2+ i sin
2)
27i3
= (27i)131 = [27 (cos
2+ i sin
2)]
131 = 3 (cos
2+2k
3+ i sin
2+2k
3)
k = 0 , 1 , 2
k = 0 3 (cos
6+ i sin
6) = 3(
3
2+
1
2 i)
k = 1 3 (cos 5
6+ i sin
5
6)
= 3 [cos (
6) + i sin (
6)] = 3 (cos
6+ i sin
6)= 3(
3
2+
1
2 i)
k = 2 3 (cos 9
6+ i sin
9
6) = 3 (cos
3
2+ i sin
3
2) = 3(0-i) = -3i
27i3
= { 33
2+
3
2 i ,
33
2+
3
2 i , -3i }
. (16-) -1 /
-16 = 16(-1) = 16 (cos + i sin )
164
= [16 (cos + i sin )]141 = 2 (cos
+2k
4+ i sin
+2k
4)
k = 0 , 1 , 2 , 3
k = 0 164
= 2(cos
4+ i sin
4) = 2(
1
2 +
1
2 i) = 2 + 2 i
k = 1 164
= 2(cos3
4+ i sin
3
4)
3
4
= 2 [cos (
4) + i sin (
4)]= 2 (cos
4+ i sin
4)
= 2(1
2 +
1
2 i) = 2 + 2 i
5
6
-
) ( [ 9 1 ]
80087450770 / 40
k = 2 164
= 2(cos5
4+ i sin
5
4)
5
4
= 2 [cos ( +
4) + i sin ( +
4)]
= 2 (cos
4 i sin
4) = 2(
1
2 -
1
2 i) = 2 2 i
k = 3 164
= 2(cos7
4+ i sin
7
4)
7
4
= 2 [cos (2
4) + i sin (2
4)]
= 2 (cos
4 i sin
4)= 2(
1
2 -
1
2 i) = 2 2 i
164
= {(2 + 2 i) , (2 2 i)} . (64i-) -0
/
-64i = 14(-i) = 64 (cos3
2+ i sin
3
2)
64i 6
= [64 (cos3
2+ i sin
3
2)]
161 = 2 (cos
3
2+2k
6+ i sin
3
2+2k
6)
k = 0 , 1 , 2 , 3 , 4 , 5
k = 0 64i6
= 2(cos3
12+ i sin
3
12) = 2(cos
4+ i sin
4)
= 2( 1
2 +
1
2 i) = 2 + 2 i
k = 1 2(cos7
12+ i sin
7
12)
k = 2 2(cos11
12+ i sin
11
12)
k = 3 2(cos15
12+ i sin
15
12) = 2(cos
4 i sin
4)= 2(
1
2
1
2 i) = 2 2 i
k = 4 2(cos19
12+ i sin
19
12)
k = 5 2(cos23
12+ i sin
23
12)
-
) (
80087450770 / 40
x,y R -8y
1+i=
x2+4
x+2i
3w9n) -2 +5
w5+
4
w4 ) (. n Z 6(
-71+3i
1+3z ,
121
: -1
a) 7+iw+iw2
1iwiw2 b) 23i + (1 + w4)6 (1 + w5)3
c) 5
3( 81 )
= a + bi a , b R -274i
2+i2a bi
: x , y R -3
a)(1i
1+i)2
+ 1
x+yi= 1 + i b) (2+xi)(-x+i)=
9y2+49
3y+7i
c) (y + 3)2 = x327i
x23xi9 d)
iw2+i
w2= xw + ywi
e) (1i
1+i) x + (1 + 3i)y = (1 i)(1 + 3i)
-44x+i
3+2i ,
yi
1i .x , y R ,
i 3 : -5 = 1 - 3x
.wy 2x2w +2, ix = 2+ ,3 i -y = 2 3 -6 . (i-) -7 : -8
a) (cos10 + i sin 10)10(cos20 + i sin 20)6
(cos30 + i sin 30)7(cos25 + i sin 25)4 = i
b) (1
1+3w2
1
1+3w4)2=
27
49
R x x 5x +10 0 = 1 + : x+1=0 2x+ -9
i = 0 x 2ix -11
( : -117
w2+ 2.(7 + 9w2+ 2)
6
wk = (2w + : ki = 0 5x
: x3 -12 3 i = 0
C : 3n+2+ w 1-3nC = w ,N n C -13
(6 : -14
6sin i+
6x = (cos z :
|x|+ 2w xw xz =
(6 : -154
w4+
5
w5k = (3 + + ki = 0 3x
2887 (8i- ) -16
-
) (
80087450770 / 47
x,y -17 y
1+i=
x2+2
x+2i 2887 .
z1z 2887+2 i4 + 3= 1z ,+ 2i 5= 2 z -18
) : -195w2i 1
5 + i w)6
= 2887 1
-21(1i)11
64 2887 .
) 3i - = 4 2c 4i - = 7 1c -21
)= (
)
2884
-22w
3w2 ,
w2
3w 2884
8 2884 (yi = (3x + i)(x+2i+7 y x -23
cos] -247
12 + sin
7
12]3
8 2884
-
) (
80087450770 / 58
-
2015 / 2016
-
( ) [ 1 2 ]
80087430770 / 35
: ( )
: :
.(Circle)
) Parabola.)
(Ellipse.)
(.Hyperbola )
:
by + c = 0ax + , )y1x,1( : 2] [1 y1(x,1( ,
. (e) ax + by + c = 0 :
y1(x,1( -1 . Focus
ax + by + c = 0 -2 .Directrix
(e) -5Eccentricity.
y1(x,1( (x,y) :
.
S: (x y1(x,1( (x,y) x1)2 + (y y1)
2 S =
= D: D ax+by+c=0 (x,y) |ax + by + c |
a2 + b2
: e
e = (xx1)
2+(yy1)2
|ax + by + c |
a2 + b2
(x x1)2 + (y y1)2 = e. |ax + by + c |
a2 + b2
:
(x x1)2 + (y y1)
2 = e2.(ax + by + c)2
a2 + b2
.
/ Parabola)) e = 1)) Ellipse)) e < 1))
Parabola)) e > 1))
-
( ) [ 2 2 ]
80087430770 / 35
: (Parabola) 2 ] [ 2 (M(x,y :
F(P,0) P > 0 D .
) O : X (
( , : )
MF = MQ MF
MQ= e = 1
axis) (x , (F(p,0
Q(-p,y) D ,M(x,y)
O(0,0) :
:
MF = MQ (x p)2 + (y 0)2 = (x + p)2 + (y y)2 :
(x p)2 + y2 = (x + p)2 x2 2xp + p2 + y2 = x2 + 2xp + p2 y2 = x2 + 2xp + p2 - x2 + 2xp - p2 y2 = 4px >
x = -p (F(p,0 ) (
) : (
:
y2 = -4px >
) x = p (F(-p,0 (
Y
X
F(p,0)
M(x,y) Q(-p,y)
O
Y
X F(p,0)
M(x,y) Q(-p,y)
O
Y
X F(-p,0)
M(x,y) Q(p,y)
O
-
( ) [ 2 2 ]
80087430770 / 33
= p ,0 > x : 2y < 0 : 4px 2y =/ > . y y
.8x 2y = /1 : /
. -1 : -2
y2 = 8x y2 = 4px
4p = 8 p = 2 x = -p x = -2 F(p,0) = F(2,0)
- : /2 x = -1 (F(1,0 (1 . (F(3,0 (2 . 2x-6=0 (5
/ -: x = -1 (F(1,0 (1
. x = -1 1/
F(1,0) p = 1 y2 = 4px y2 = 4 x
x > 0 - :/ , 2- 2 1
.(2-,1) (1,2)-,x) (x,y) - :
y) .
-: (F(3,0 (2 3/
.
F(p,0) = (3,0) p = 3 y2 = 4px y2 = 4 (3) x y2 = 12x
-: 2x - 6 = 0 (52x - 6 = 0 2x = 6 x = 3
.
y2 = -4px y2 = -4(3)x y2 = -12x
16 4 1 x
8 5 2 y
Y
X
F(1,0)
(1,2)
(1,-2)
O
(-1,0)
x = -1
-
( ) [ 2 2 ]
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, . 4x-= 2y /3 /
y2 = -4x y2 = -4px
-4p = -4 p = 1 x = p x = 1 F(-p,0) = F(-1,0)
/ .y x
. (3,0) /4 /
. (M(x,y
F(3,0) .
Q(3,0) M D .
:
MF = MQ (x 3)2
+ y2 = (x + 3)2
:
(x 3)2
+ y2 = (x + 3)2 x2 23x + 5 + y2 = x2 + 23x + 5
y2 = x2 + 23x + 5 - x2 + 23x - 3 y2 = 23x + 23x y2 = 43x
axis) (y , (F(0,p
(M(x,y, D (Q(x,-p
: (O(0,0
:
MF = MQ (x 0)2 + (y p)2 = (x x)2 + (y + p)2 :
x2 + (y p)2 = (y + p)2 x2 + y2 2yp + p2 = y2 + 2yp + p2 x2 = y2 + 2yp + p2 - y2 + 2yp - p2 x2 = 4py >
-4 -1 x
4 2 y
Y
X
F(-1,0)
(-1,2)
(-1,-2)
O
(1,0)
x = 1
Y
X F(3,0)
M(x,y) Q(-3,y)
O
Y
X
F(0,p) M(x,y)
Q(x,-p)
O
-
( ) [ 2 2 ]
80087430770 / 35
y = -p (F(0,p
()
y = 1
4px2
x R y y x.
) : (
:
x2 = -4py >
F(0,-p) y = p
, . y = -2 (F(0,2 /5 /
F(0,2) = F(0,p) p = 2
x2 = 4py
x2 = 8y
x y
4 2
y/ .x
, x y > 0/ . (4,2-) (4,2)
, (x,y-), (x,y) : .
.24y = 0 23x /6
/
3x2 24y = 0 3x2 = 24y x2 = 24
3 y
x2 = 8y
x2 = 8y x2 = 4py
4p = 8 p = 2 y = -p y = -2 F(0,p) = F(0,2)
Y
X
F(0,2) (-4,2) (4,2)
O
Y
X
F(0,-p)
M(x,y)
Q(x,p)
O
-
( ) [ 2 2 ]
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: /7 . (0,5) -1 . y = 7 -2
/ (1
F(0,5) = F(0,p) p = 5
x2 = 4py
x2 = 4(5) y x2 = 20y . y = 7 (2 .
y = 7 p = 7
x2 = -4py
x2 = -4(7)y x2 = -28y
(3,0) , , /8 . (M(x,y /
F(0, . (3
Q(x, . D M (3
:
MF = MQ (x 0)2 + (y 3)2
= (x x)2 + (y + 3)2
:
x2 + (y 3)2
= (y + 3)2 x2 + y2 23y + 3 = y2 + 23y + 3
x2 = y2 + 23y + 3 - y2 + 23y - 3 x2 = 43y
- : /9 (2,4),(4-,2) (1 (2,4),(4 ,2-) (2
/ (2,4),(4-,2) (1
(2,4),(4-,2)/
.
y2 = 4px 42 = 4 . p . 2 16 = 8 . p p = 2 y2 = 4px = 4 . 2 . x y2 = 8x
(2,4),(4 ,2-) (2
Y
X
F(0, 3) M(x,y)
Q(x,- 3)
O
-
( ) [ 2 2 ]
80087430770 / 35
(2,4),(4 ,2-) / .
x2 = 4py 22 = 4 . p . 4 4 = 16 . p p = 1
4
x2 = 4( 14 ) y x2 = y
.(5-,3) /11 - :/ :/
y = -5 .
p = 5
x2 = 4py x2 = 20y
: /x = 3
. p = 3
y2 = -4px y2 = -12x
, /11 (y,2) ,
, 18 .y ,
/
MF+MQ = 18 MF = MQ MF = 9 , MQ = 9
MF = (p 2)2 + (y 0)2 = 9
: MQ 4px 2y = p,y
MQ = (2 + p)2 + (y y)2 = 9 4 + 4p + p2 = 9 4 + 4p + p2= 81
p2 + 4p 77 = 0 (p + 11)(p - 7) = 0
p + 11 = 0 p = -11
p 7 = 0 p = 7
y2 = 4px y2 = 28x
: Q y
y2 = 28x y2 = 28(2) = 56 y = 56
Y
X
F(-p,0)
(3,-5)
O
Y
X
F(0,p)
(3,-5)
O
D
Y
X F(p,0)
M(2,y) Q(-p,y)
O
-
( ) [ 2 2 ]
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:p > 0
y- axis y=-p (0,p) =4py 2x
y- axis y=p (0,-p) 4py-= 2x
x- axis x=-p (p,0) x=4p 2y
x- axis x=p (-p, 0) x4p-= 2y
-
( ) [ 3 2 ]
80087430770 / 51
: 2 ] [ 3 .(h,k)
4px 2y = /
. (O(0,0
(O(h,k h) = 4p(x 2k) (y - :
(O(h,k
.
(O(0,0) O(h , k (F(p,0) F(p+h , k x = -p x = -p + h y = 0 y = k
P
F O Q = p+h |p = |Qh
:
. (x 2(y+1)4 =-(2 /12
/
(y + 1)2 = 4(x - 2)
(y k)2 = 4p(x h)
h = 2 , k = -1 O(h , k) = (2, -1) 4p = 4 p = 1 F(p+h,k) = F(1+2 ,-1) = F(3 ,-1) y = k y = -1 x = -p + h x = -1 + 2 x = 1
Y
F(Q,K)
X
Y
(h, k)
X
(y k)2 = -4p(x h)
(F(-p+h , k x = p + h
y = k
A
Y
X
F(Q,K) (0,0)
x = -p
B
Y
X
F(Q,K)
x = -p + h
(h, k)
-
( ) [ 3 2 ]
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4py 2x = /
. (O(0,0
(O(h,k k) = 4p(y 2h) (x - :
(O(h,k .
(O(0,0) O(h , k (F(0,p) F(h,p+k y = -p y = k - p x = 0 x = h
Q = p+k |p = |Qk
) (:
4x 2y = x + /13
) ( 2x / :5 x
y + 4 = x2 + 4x + 4
y + 4 = (x + 2)2
(x + 2)2 = y + 4
(x h)2 = 4p(y k)
h = -2 , k = -4
(x h)2 = -4p(y k)
(F(h , -p+k y = p + k
x = h
y = k+p
Y
X
F(h,k-p)
(h, k)
A Y
X
F(0,p)
(0,0)
y = -p
Y
X
F(h,p+K)
(h, k)
B
-
( ) [ 3 2 ]
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4p = 1 p = 1
4
O(h , k) = (-2 , -4)
F(h, p+k) F (-2 , 1
4 4) F (-2 , 3 3
4 (
y = k p y = -4 - 1
4 y = -4 1
4
.(3-,3) (5-,3) /14 x = 3 /
(x h)2 = 4p(y k) :
F(3,-3) = (h , p+k) h = 3 , k = -5 p + k = -3 p 5 = -3 p = 2
(x 3)2 = 4(2)(y + 5)
(x 3)2 = 8(y + 5)
2 - 1
: (1 . (5,0) .
/F(5,0) p = 5
y2 = 4px y2 = 20x x = -p x = -5
. (4-,0) . (F(0,-4 /
p = 4
x2 = -4py x2 = -16y y = p y = 4
4 0 x -1 0 y
1 0 x
2 5 0 y
Y
X
F(0,-4)
y = 4
O
D
Y
X
F(5,0) O
D
x = -5
Y
X
F(3,-3)
(3 , 5)
-
( ) [ 3 2 ]
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. (2,0) . (F(0, 2 /
p = 2
x2 = 4py x2 = 52y
y = -p y = -2
. 4y 3 = 0 . /
4y 3 = 0 y = 3
4
F(0 , 3
4) p =
3
4
x2 = -4py x2 = -5y
3 0 x
-1 0 y
. (1,2) /
. /
y2 = 4px
22 = 4p(1) p (1,2)
p = 1 F(p,0) = (1 , 0) x = -p x = -1
y2 = 4x
1 0 x
2 0 y
: (2a) x2 = 4y
/
x2 = 4y
x2 = 4py
4p = 4 p = 1 F(0,p) = (0,1) y = -p y = -1 (0 , 0)x = 0 ) )
8 0 x
2 0 y
Y
X
F(1,0) O
D
x = -1
Y
X
F(0,- 3
4)
y = 3
4
O
D
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( ) [ 3 2 ]
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b) 2x + 16y2 = 0
/
2x + 16y2 = 0 16y2 = -2x y2 = 2
16 x
y2 = 1
8 x
y2 = -4px
-4p = 1
8 4p =
1
8 p =
1
32
c) y2 = -4(x - 2)
/
y2 = -4(x - 2)
(y 0)2 = -4(x - 2)
(y k)2 = -4p(x - h)
h = 2 , k = 0 -4 = -4p p = 1
d) (x - 1)2 = 8(y 1)
/
(x - 1)2 = 8(y 1)
(x h)2 = 4p(y - k)
h = 1 , k = 1 4p = 8 p = 2
e) y2 + 4y + 2x = -6
/
y2 + 4y = -2x 6
: y
y2 + 4y + 4 = -2x 6 + 4
(y + 2)2 = -2x 2
(y + 2)2 = -2(x + 1)
(y k)2 = -4p(x - h)
h = -1 , k = -2
-4p = -2 p = 2
4 =
1
2
F(1
32 (0 ,
x = 1
32
(0 , 0)y = 0 ) )
O(h,k) = (2,0) F(-p+h k) = (-1+2 , 0) = (1,0) x = p + h = 1+2 = 3 x = 3 y = k y = 0
O(h,k) = (1,1) F(h ,p+k) = (1 , 2+1) = (1, 3) y = -p + k = -2+1= -1 y = -1 x = h x = 1
O(h,k) = (-1,-2)
F(-p+h , k) =(12 -1, -2) =(
32 (2-,
x = p + h = 1
2 1= 1
2 x =
12
y = k y = -2
-
( ) [ 3 2 ]
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f) x2 + 6x-y = 0
: x 6x = y 2 x + /
x2 + 6x + 9 = y + 9
(x + 3)2 = (y + 9)
(x h)2 = 4p(y - k)
h = -3 , k = -9
4p = 1 p = 1
4
. (2,5),(5-,2) (3
(2,5),(5-,2) / ( )
y2 = 4px : p
52 = 4p(2) 25 = 8p p = 25
8
y2 = 4( 25
8)x p
y2 = 25
2 x
. (5-,2-),(5-,2) / (5-,2-),(5-,2) /
) (
x2 = -4py : p
22 = -4p(-5) 4 = 20p p = 1
5
x2 = -4( 1
5) y p
x2 = 4 5
y
, (3,4-) (4 .
- : , (3,4-) / / :
x = -3 p = 3
y2 = 4px
y2 = 12x
Y
X
F(3,0) O
D
x = -3
O(h,k) = (-3,-9)
F(h ,p+k) = (-3, 1
4 -9) = (-3,-
35
4 (
y = -p + k = 14 - 9 y =
374
x = h x = -3
-
( ) [ 3 2 ]
80087430770 / 55
:/ y = 4 p = 4
x2 = -4py
x2 = -16y
. A, (1,2) 8y = 0 2Ax + (5 : (1,2) /
A(1)2 + 8(2) = 0
A + 16 = 0 A = -16
-16x2 + 8y = 0 16x2 = 8y
x2 = 1
2 y
x2 = 4py
4p = 1
2 p =
1
8
F(0 , 1
8 (
y = - 1
8
: , (6 . (7,0) .
/F(7 , 0) x = -7 M(x,y) , Q(-p,y) = (-7,y)
:
MF = MQ (x 7)2 + y2 = (x + 7)2 + 0 (x 7)2 + y2 = (x + 2(7x2 14x + 49 + y2 = x2 + 14x + 49 y2 = 28x
1 0 x 2 0 y
Y
X
F(0,-4)
y = 4
O
D
Y
X
F(0, 1
8)
(0,0)
D y = -p
(1,2) (-1,2)
-
( ) [ 3 2 ]
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. y = 3 .y = 3
F(0,- 3) M(x,y) , Q(x,p) = (x, 3) MF = MQ
x2 + (y + 3)2
= 0 + (y 3)2
x2 + (y + 3)2
= (y 3)2
x2 + y2 + 23y + 3 = y2 - 23y + 3 x2 = -43y
Y
X
F(0,- 3)
M(x,y)
Q(x, 3)
O
D y = 3
-
(Ellipse) ( ) [ 4 2 ]
80087430770 / 55
(:Ellipse) 2 ] 4 [ -:
2a)( = , Center ) (
Focal axis ) ( .
:
: (P(x,y
PF1 + PF2 = 2a
(x c)2 + (y + 0)2 + (x + c)2 + (y + 0)2 = 2a
:
+
=
- : c,0)-(2(c,0) , F1F -1 a-(2,0) , Va(1V(0, -2 ((b) , (0,-b,0 )( : -5 .2a (Major axis) -5 .2b (Minor axis) -3
F1F2 -5 . 2c
a > b a > c 2b - 2= a 2c c 2= b 2a +2 a , b , c -5
. a,b,c
= e -8c
a e < 1 1
:
: (P(x,y
F2
F1
(0,-c)
(0,c)
V2(0, -a)
M1(b,0)
M2
(V1(6,a P(x,y)
(-b,0)
F2 F1
(-c,0) (c,0)
V2
(-a,0)
M1(0,b)
(M2(0,-b
V1
(a,0)
P(x,y)
y
x
-
(Ellipse) ( ) [ 4 2 ]
80087430770 / 56
PF1 + PF2 = 2a
x2 + (y c)2 + x2 + (y + c)2 = 2a
:
+
=
- : c)-0,(2c) , F0,(1F -1 a-0,(2) , V0,a(1V( -2 ((b,0) , (-b,0 )( : -5 .2a (Major axis) -5 .2b (Minor axis) -3
F1F2 -5 . 2c
a,b,c a > b a > c 2b - 2= a 2c c 2= b 2a +2 a , b , c -5 .
= e -8c
a e < 1 1
-:
Area (A) A = a b : -1
P) Perimeter : P = 2) -2a2+b2
2
e -5 .y x -5 . -3 : a , b -5
a. = 1 .
b. 2 x = 1 .
c. 2 y = 1 .
. 2y 2x -5
. 2y 2x -8 )( ) ( -5
.b a ,
2 = a
= 2 b
= 2 c
a , b , c 2+ c 2= b 2a
2b - 2= a 2c
x2
a2+
y2
b2= 1
x2
b2+
y2
a2= 1
a)-(0,2(0,a) , V1V a,0)-(2(a,0) , V1V
c)-(0,2(0,c) , F1F c,0)-(2(c,0) , F1F
((b,0) , (-b,0 ((b) , (0,-b,0
-
(Ellipse) ( ) [ 4 2 ]
80087430770 / 51
. /15
1) x2
25+
y2
16= 1
2y 2x /
x2
25+
y2
16= 1
x2
a2+
y2
b2= 1
a2 = 25 a = 5 2a = 10 b2 = 16 b = 4 2b = 8
a2 = b2 + c2 25 = 16 + c2 c2 = 9 c = 3
F1(3,0) , F2(-3,0)
V1(5,0) , V2(-5,0)
e = c
a =
3
5 1 >
2) 4x2 + 3y2 = 4
3
/ 4
3
5x2 + 9
4y2 = 1
x2 1
3
+ y2 4
9
= 1
2y 2x :
x2 1
3
+ y2 4
9
= 1
x2
b2+
y2
a2= 1
a2 = 4
9 a =
2
3 2a =
4
3
b2 = 1
3 b =
1
3 2b =
2
3
c2 = a2 - b2 c2 = 4
9
1
3 c2 =
1
9 c =
1
3
F(0, 1
3 (
V(0, 2
3 (
e = c
a =
1
3
2
3 =
1
2 1 >
-
(Ellipse) ( ) [ 4 2 ]
80087430770 / 52
3) 4x2 + y2 = 4 5 /
x2 + 1
4 y2 = 1
x2
1 +
y2
4 = 1
:
x2
1 +
y2
4 = 1
x2
b2+
y2
a2= 1
a2 = 4 a = 2 2a = 4 b2 = 1 b = 1 2b = 2
c2 = a2 b2 c2 = 4 1 c2 = 3 c = 3
F(0, 3) V(0, 2)
e = c
a =
3
2 1 >
, /16 , 12 8
.
/
2a = 12 a = 6 2b = 8 b = 4 x2
b2+
y2
a2= 1
x2
16+
y2
36= 1
c2 = a2 - b2 c2 = 55 16 = 20 c = 20 = 25
2c = 45 A = a.b. = 6(4) = 24 unit2
P = 2a2+b2
2 = 2
36+16
2 = 2
52
2 = 2 26 unit
F3(1F , (2,0)-3(0, /17 . 16
/F1(3,0) , F2(-3,0) c = 3 2a = 10 a = 5
a2 = b2 + c2 25 = b2 + 9 b2 = 16 b = 4 x2
a2+
y2
b2= 1
x2
25+
y2
16= 1
, /18 . 2 5
-
(Ellipse) ( ) [ 4 2 ]
80087430770 / 55
/2c = 6 c = 3 2a 2b = 2 a b = 1 a = b + 1 .
a2 = b2 + c2 .. :
(b+1)2 = b2 + 9 b2 + 2b + 1 = b2 + 9 2b + 1 = 9 2b = 8 b = 4 a = b + 1 a = 4 + 1 = 5 x2
25+
y2
16= 1
/19
(4,0-) , (4,0) 2
5.
(4,0-) , (4,0) / .
b = 4 x2
b2+
y2
a2= 1
: 2b
2a=
2
5
8
2a=
2
5 4a = 40 a = 10
x2
16+
y2
100= 1
/21 . 3 , 7
/2a = 3 + 7 = 10 a = 5
c = 5 3 = 2
a2 = b2 + c2 25 = b2 + 4 b2 = 21
x2
25+
y2
21= 1
.k, (3,0) , , 4y 2kx +2 36 = /21
55 /
kx2 + 4y2 = 36 x2
36
k
+y2
9= 1
a2 = 36
k , b2 = 9 , c2 = 5
a2 = b2 + c2 36
k = 9 + 5 = 18 k =
36
18 = 2
F1
V2
(-a,0)
V1
(a,0)
3 7
c a
-
(Ellipse) ( ) [ 4 2 ]
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5 = F2(1F , (2,0)-2(0, , /22 .
(P(x,y /PF1 + PF2 = 2a
(x 2)2 + (y + 0)2 + (x + 2)2 + (y 0)2 = 6
(x 2)2 + y2 = 6 - (x + 2)2 + y2
(x 2)2 + y2 = 36 - 12(x + 2)2 + y2+ (x + 2)2 + y2
x2 4x +4 +y2 = 36 - 12(x + 2)2 + y2 + x2 +4x +4 +y2
4x = 36 - 12(x + 2)2 + y2 + 4x 12(x + 2)2 + y2 = 36 + 8x
3(x + 2)2 + y2 = 9 + 2x 5
9((x + 2)2 + y2) = 81 + 36x + 4x2 9(x2 + 4x + 4 + y2) = 81 + 36x + 4x2
9x2 + 36x + 36 + 9y2 = 81 + 36x + 4x2 9x2 + 36 + 9y2 = 81 + 4x2
5x2 + 9y2 = 81 36 5x2 + 9y2 = 45 53 x2
9+
y2
5= 1
, /23 12x = 0 2y . 16
: /y2 12x = 0 y2 = 12x y2 = 4px 4p = 12 p = 3
. (F(3,0 2F ,(3,0)1F)-(3,0
c = 3 2b = 10 b = 5 a2 = b2 + c2 a2 = 25 + 9 = 34 x2
34+
y2
25= 1
:
V 1V ,2 -1
M1M ,2 -2
M 2, V 1, M 1V ,2 -5
F 1F ,2 -5
-
( ) [ 5 2 ]
80087430770 / 53
, : 2 ] [ 5 (h , k)
:
(h,k) h (O(0,0
- : k
(xh)2
a2+
(yk)2
b2= 1
c , b , a 2c + 2b = 2a a > b , c
O(0 , 0) (h , k) 0 ) , c-(20) , F , (c1F
h , k) + c-(2h , k) , + (c1 0 ) , a-(20) , V , (a1V
h , k) + a-(2h , k) , + (a1 b)- , (02b) , M , (01M
k) + b-(h,2k) , + b , (h1
2 a , y = k
2 b , x = h
(h,k)
h (O(0,0 - : k
(xh)2
b2+
(yk)2
a2= 1
a,b,c 2+c2=b2a a > b,c
O(0 , 0) (h , k) )c ,0 (2) , Fc ,0 (1F
k) + c- (h ,2k) , + c(h , 1 )a- ,0(2) , Va ,0 (1V
k) + a- (h ,2k) , + a(h , 1 0( ,b -(2) , M0 ,b (1M
k), b- (h2k) , ,b + (h1
2a , x = h
2b , y = k
F2 F1
(-c+h ,k) (c+h ,k)
M1(h ,-b+k)
V1 (a+h ,k)
V2
(-a+h ,k)
M2(h ,b+k)
F2
F1
(h ,-a+k)
(h ,c+k)
M1 (b+h ,k)
V1 (h ,a+k)
V2
M2 (-b+h ,k)
(h,-c+k)
-
( ) [ 5 2 ]
80087430770 / 55
: /24(x2)2
9+
(y1)2
25= .e, 1
, /(x2)2
9+
(y1)2
25= 1
(xh)2
b2+
(yk)2
a2= 1
b2 = 9 b = 3 a2 = 25 a = 5 h = 2 , k = 1
(h , k) = (2,1) a2 = b2 + c2 25 = 9 + c2 c2 = 25 9 = 16 c = 4
2 - 2 (1
:
a) x2 + 2y2 = 1
: /
x2 + 2y2 = 1
x2
1+
y2
1
2
= 1
x2
a2+
y2
b2= 1
a2 = 1 a = 1
b2 = 1
2 b =
1
2
a2 = b2 + c2 1 = 1
2 + c2
c2 = 1
2 c =
1
2
b) 9x2 + 13y2 = 117
: /
9x2 + 13y2 = 117 9x2
117+
13y2
117= 1
x2
13+
y2
9= 1
x2
a2+
y2
b2= 1
:
2y 2x (h , k) = (6,6)
2y 2x
(h , k) = (6,6)
F(1
2 (0 ,
V(1, 0)
M(0, 1
2 (
2a = 2(1) = 2unit y = 0
2b = 2(1
2) = 2 unit
x = 0
e = c
a =
1
2
1 =
1
2 1 >
2a = 10 unit : x = 2 : 2b = 6 unit : y = 1 :
:
F1(h , c+k) = (2, 4+1) = (2,5) F2(h , -c+k) = (2, -4+1) = (2,-3)
:
V1(h , a+k) = (2,5+1) = (2,6) V2(h , -a+k) = (2,-5+1) = (2,-4)
e = c
a =
4
5 1 >
-
( ) [ 5 2 ]
80087430770 / 55
a2 = 15 a = 13
b2 = 9 b = 3 c2 = a2 - b2 c2 =13 -9 = 4 c = 2 F(2, 0) V(13, 0) M(0, 3) 2a = 2(13) = 213 unit y = 0 2b = 2(3) = 6 unit x = 0
e = c
a =
2
13 1 >