I Vay CO hai can bac hai (dang dai so) cu aw la:
Phuong trinh da cho tuong duong voi:
({1 - 2i) - ( 3 = i))z = i l i « (-2 - i)z = 1 ^
1 + 1 2
1 7 * o z = ị Vay diem bieu dien ciia z la M o z = ị Vay diem bieu dien ciia z la M
10 10 JÓlO,
Phuong trinh da cho tuong duong voi:
(z - 1)2 - (x - i)2 = 0 <=> (z - i)(z - 2 + i) = 0<::>z = ihayz = 2 - i
Do do: Z I + Z2 = + | 2 - i | = N/S
Bai 20.
a) Cho so phuc thoa man: (2 + i)z + ^^'^ = 7 + 8ị T i m m o d u n ciia sóphuc: w = z + 1 + i
b) Giai p h u o n g t r i n h : i ? + 3(1 + i)z + 5i tren tap hop so phuc. G i a i
a) Phuong trinh t u o n g d u o n g : (2 + i)z = 4 + 7i <=>z = 3 + 2i=:i>w = 4 + 3 i = > | w | = N/4^+3^ = 5 b) A = - 2 i = (1 - i)^ d o d o cac nghiem ciia p h u o n g t r i n h la:
z = l ( ^ i ^ ^ = - l - 2 i , ^ ^ - ( 3 . i ) - ( l - i ) _ ^ _ .
2
Bai 21. Goi z,, z, la nghiem cua p h u o n g trinh: iz - 4 = 0. Hay viét dang l u g n g giac ciia z,, Z2
Giai:
Ta CO A = 4 d o d o z, = 1 + i ; Z2 = - 1 + i Dang l u o n g giac cua z^ la:
7t . . 7t , , , „ , 27t . . 27t .
z, = 2 ( c o s - + i s m - ) va cua z, la: Z-,-2(cos — + ism — )
' 3 3 - ' 3 3
Bai 22. Cho so phuc z thoa man ^^^^'^ = 2 - ị T i m m o d u n cua so phiic: w = 1 + z + z + i G i a i D a t z = a + b i ( z ^ - l ) T a c o : ^ ^ ^ ^ ^ = 2 - i » (3a - b - 2) + (a - 7b + 6)i = 0 z + i 3 a - b - 2 = 0 ^ ^ , <::>a = l , b = l =>Z = 1 + 1 a - 7 b + 6 = 0 w =•! + z + z 2 = 2 + 3i => I w I = I 2 + 3i I = 7T3 Bai 23. a) T i m so phuc z thoa m a n z^ = I z h + z
b) T i m m o d u n ciia so phuc z thoa man: (2z - 1 ) ( 1 + i) + ( z + 1 ) ( 1 - i) = 2 - 2i
b) Ta c6: 1 + i N/S = 2(cos — + isin —) => (1 + i VB ý' = 8(cos7t + isinn) 3 3
z = „ / T , 371 . . 3:1, —^ — = 2 N/2 (cos - + isin - ) = 2 + 2 i 8(cos7t + i s i n 7 i ) ~ fZ, K . . 7 t , ^ ^. 2v2(cos + i s i n )
4 4
Bai 25. T i m so p h i i c z biet z - (2 + 2i) z = 1 - 9i
G i a i
Dat z = a + bi, ta c6: z - (2 + 3i) z = 1 - 9i <=> a + bi - (2 + 3i)(a - bi) = 1 - 9i
-a - 3b = 1
<r> - a - 3b + (3b - 3a)i = 1 - 9i o
3 b - 3 a = - 9
<=>a = 2 , b = - l o z = 2 - i . * *, Bai 26: Goi cac nghiem ciia p h u o n g trinh z^ + 1 = 0 la z,, Z j , Z3.
H a y tinh S= +zl+ Z3 G i a i z''+ 1 = 0 o (z + l ) ( z 2 - z + 1) = 0. Suy ra I N / 3 . 1 z, = - 1 , z, = — + — 1; z, = 1 2 2 2 ^ 2 2 S = (-1)'^ + ( c o s - + i s i n - + ( c o s ( - - ) + isin(--))'^ ; 3 3 3 3 , 57t . . 571 , 571 , . . , 57t ,
= - 1 + cos — + isin — + cos( ) + isin( ) 3 3 3 ' ^ 3 S = - ] + 2 C O S - - = - 1 + 1 = 0
3
[ Bai 27: Cho z la so phuc khac 1, goi A, M , M ' Ian l u o t la cac d i e m bieu dien Ian lugt ciia: 1, z, z^. Xac djnh cac diem M sao cho tam giac A M M ' la tam giac deụ
G i a i A A M M ' la tam giac deu <=> A M = A M ' = M M '
Z - 1 = I Z2 - 1 I = Z 2 - Z
M a t khac: I z^ - 1 | = | (z - l ) ( z + 1) = I z - 1 | z + 1 va I z^ - z I = I z(z - 1 ) 1 = |z| . | z - l |
vi z 7t 1 => z - 1 ^ 0 d o d o A A M M ' deu <=> 1 = z + 1 = I z z I = 1 o M 6 tren d u o n g tron tam O ban k i n h R = 1 z I = 1 o M 6 tren d u o n g tron tam O ban k i n h R = 1
Cam nang Ivy^n thi DH - Nguyen ham - Tit h pluhi - So phírc - Trdn Bd Ha
G i a i
a) Dat z = a + bi, ta c6: z^ = - b^ + 2abi
I z h + z = a2 + b^ + a - bi, do do z2 = I z h + z
a^-b^ =â + b^+a a = -2b^
2ab = -b <=> { b(2a +1) = 0 <=> a = 0, b = 0 hay a = - - , b = - h a y a = - - , b = - - y 2 2 ^ 2 2 W ' U - u - n 1 1 . 1 1 . Vay CO ba so phuc: z = 0, z = - • ^ 2 ' ' ^ ^ ~ 2 " 2 * b) Dat z = a + bi, ta c6: ( 2 z - i ) ( 1 + i ) + ( z +1)(1 - i ) = 2 - 2 i » 3 a - 3 b + (a + b - 2 ) i = 2 - 2 i <=> 3a - 3b = 1 a + b - 2 = - 2 1 a = — 3 b = - - 3 D o do I z I = yjâ + h^ = Bai 24.
a) T i m sóphuc z thoa man: z - - 1 =^0 b) T i m phan thuc phan ao ciia so phuc: z =
G i a i a ) D a t z = a + bi, ta c6: z - ^ " ^ ' ^ - 1 = 0 ^ l + i>/3l 1 + i o a _ b i + ^ ± 1^ - 1 = 0 o a 2 + b 2- 5 - i N / 3 - a - b i = 0 a + b i <=> (â + b^ - a - 5) - (b + N/3 ) i = 0 o o a = - 1 , b = - >/3 hoac a = 2, b = - Vs Vay z = - 1 - Vi i h o l e z = 2 - Vs i a ^ + b ^ - a - 5 - 0 b + N/3=0 347
IT Cam nang luyCn ihi DH - Nguyen ham - Tich phan - So phírc - Tmyy Bd Ha
Vay cac diem can t i m la giao diem cua d u a n g th3ng ~ ^ v o i d u o n g tron:
x2 + y2 = l ^ M i ( - - , + — ) , M 2 ( - - , - — ) ,
y ' 2 2 2 2 ?
Bai 28: Vai moi só phuc z ^ 1 dat w = — — , voi z = x + iy z + 1