1. D i n h nghia: mot so phuc la mpt bieu thuc dang: a + bi trong do a,b la cac so thuc va so i thoa man: i^ = - 1 . Ky hi^u so phvic: Z = a + b i ; a goi la phaV, thuc va so i thoa man: i^ = - 1 . Ky hi^u so phvic: Z = a + b i ; a goi la phaV, thuc, b goi la phan aọ
CM y:
+ M o i so thuc a duoc coi la mpt so phuc c6 phan ao bang 0: z = a -t- Oi = a e R + So phuc CO phan thuc b3ng 0 ggi la so ao z = 0 + bi = bi; i = 0 + l i + So phuc CO phan thuc b3ng 0 ggi la so ao z = 0 + bi = bi; i = 0 + l i
+ Cho z = a + b i v a z' = a ' + b ' i z = z <=> < a = a z = z <=> < a = a
b = b' 2. Bieu dien h i n h hpc 2. Bieu dien h i n h hpc
+ M o i so phuc Z = a + bidugc bieu dien bai diem M(a,b) va ngugc lai, ky hieu: M(z) hoac M(a + bi) hieu: M(z) hoac M(a + bi)
+ Cac diem tren Ox bieu dien cac so thuc do do true Ox ggi la trur thuc + Cac diem tren Oy bieu dien cac so ao do do true Oy ggi la true ao + Cac diem tren Oy bieu dien cac so ao do do true Oy ggi la true ao 3. So p h u c l i e n hgrp
So phuc lien hgp cua so phuc: Z a + bi (a, b € R) la so phuc Z = a - bi + l = z + l = z
+ Hai so phuc lien hgp c6 diem bieu dien tuang ung dol xung qua Ox 4. M o d u n ctia so phuc 4. M o d u n ctia so phuc
D i n h nghia: Modun ciia so phuc Z = a + bi (a, b e R) la so phuc am Vâ + b^ ducyc ky hieu la |Z ky hieu la |Z
+ Néu Z = a + bi thi: Z = O M . M la diem bieu dien cua 2 trong he true xOy I I . Phep t i n h so phuc I I . Phep t i n h so phuc
1. Phep cpng va phep t r u so phuc:
a) Tong ciia 2 so phuc: Cho Z = a + b i v a Z ' = a ' + b ' i Z + Z ' = a + á+(b + b')i (a,b,á,b'e R) Z + Z ' = a + á+(b + b')i (a,b,á,b'e R)
+ (Z + Z') + Z " = Z + ( Z ' + Z " ) voi Z , Z ' Z " e C '
z + z = z + z
Cty 'J 'NHH MTV D VVH Khang Vi^t
•A
z+o=ơz=z
Z = a + bi thi - Z = -a - bi ggi la só phuc dol ciia Z+(-Z)=0
J) Hieu ciia hai so phuc
Hieu cua hai so phuc Z va Z' la tong cua Z va - Z ' Z - Z ' = Z + ( - Z ' ) Z - Z ' = Z + ( - Z ' )
Voi Z = a + bi va Z ' = a ' + b ' i thi Z - Z ' = a - á+(b - b')i nghia h i n h hpc ciia phep cpng va phep t r u nghia h i n h hpc ciia phep cpng va phep t r u
Trong mat phang phuc M(a,b) la tieu diem bieu dien ciia sóphuc Z = a + bi ta ciing ggi u = (a,b)la bieu dien ciia Z nghia la néu M la diem bieu dien ta ciing ggi u = (a,b)la bieu dien ciia Z nghia la néu M la diem bieu dien
ciia so phiic Z = a + bi thi O M la vecto bieu dien ciia Z ' . Néu u , u ' Ian lugt la vecto bieu dien ciia Z va Z ' thi u + u ' la vecto bieu . Néu u , u ' Ian lugt la vecto bieu dien ciia Z va Z ' thi u + u ' la vecto bieu
1 ~ - • —
' dien ciia Z+Z' va u - u ' la vecto bieu dien ciia Z - Z ' 2. Phep nhan so phuc: 2. Phep nhan so phuc:
a) Dinh nghia: Cho hai so phiic Z = a + bi va Z ' = á+ b' i Z.Z' = aá-bb'+(ab + bá)i (a,b,á,b' eR) Z.Z' = aá-bb'+(ab + bá)i (a,b,á,b' eR)
i) Tinh chat: Z.Z' = Z'.Z
Z ( Z ' Z " ) = (Z.Z')Z" Z ( Z ' + Z " ) = Z.Z'+ Z.Z" Z ( Z ' + Z " ) = Z.Z'+ Z.Z"
I K e R , K . Z - k a + kbi
K
8. Phep chia so phuc khac 0
D i n h nghia: Cho so phuc Z = a + bi(a,b e R) khac 0 Z Z
So phuc Z =
â+b^ ggi la nghjch dao ciia Z
Thuong — ciia phep chia so phuc Z' cho so phuc Z khac O la tich ciia Z' voi nghjch dao ciia Z voi nghjch dao ciia Z
^ . z ' . z - = ^
Z " 2
Z' Z ' Z Z . Z ^ ^ . . Z ' . . ~, — r - = —= - Do do de tim — ta nhan mau vai Z — r - = —= - Do do de tim — ta nhan mau vai Z
' Z.Z Z
ua/w nang liiyen thi ±)tl - Nguyen ham - lich phdn - So phuc - Trdn Ba Ha
B. P h u o n g phap giai cac dang ca ban
V a n de 1: B I E U D I E N SO PHL/C D A N G H I N H H Q C
P h u o n g phap: M o i so p h u c Z= a+bi dugc b i e u d i e n v o i m p t d i e m M(a,b) hoac vecto u =(a,b) do do ap d u n g cac phep tinh vecto, toa do diem de i<hao sat t i n h chat, tap hop d i e m bieu dien cua so phuc, ket h o p cdc phep tinh l i e n quan
Bai 1: Cho cac so phuc: Z, = 3 + 2i, = 1 - 2i
a) Viet cac sóphuc doi v o i bieu dien trong chiing trong mat phang phuc b) Viet cac so phuc lien hgp va bieu dien chiing trong mat phSng phuc.
G i a i
a) Z , = 3 + 2i CO diem bieu dien M , (3,2)
So p h u c doi la: - Z , = -3 - 2i c6 diem bieu dien la '(-3, -2)
Z j = 1 - 2i CO diem bieu dien (1, -2); so phuc d o i la - Z j = - 1 + 2i c6 diem
bieu dien la: '(-1,2)
b) So p h u c lien hgp ciia Z, la Z, = 3 - 2i c6 diem bieu dien la N , (3, - 2 ) ;
Sóphuc lien hop ciia Z j la Z j = 1 + 2i c6 diem b i l u dien la '^^{'^,1)
Bai 2: T i m tap hop cac diem bieu dien ciia cac sóphuc sau: 1) Z = a + 2i k h i a t h a y doi
2) Z = a - ai k h i a thay d o i 3) Z = a + 3ai khi a thay d o i
Giki
1) Z = a + 2i CO diem bieu dien trong mat p h l n g phuc la diem M(a,2) do do khi
a thay doi thi tap hop cac diem M la d u o n g thang y = 2
2) Z = a - ai co diem bieu dien la M(a, -a) do do k h i a thay doi thi tap hgp cac d i e m M la d u o n g thang y = - x hay x + y = 0
3) Z = a + 3ai c6 diem bieu dien ciia so phiic la M(a,3a) do do tap hop cac d i e m M la d u o n g thSng y = 3x
Bai 3: Cho A la diem bieu dien ciia so phuc Z = 1 - 2 i . Goi M , Ian l u g t la
d i e m bieu dien ciia cac s o p h i i c Z-^.Z^. Chiing m i n h rang dieu Kien de
A A M j M j cantai M , la: Z , - l + 2i Z , - Z , G i a i
A A M ^ M j cantai M , la M , A = M , M 2
M i A = Z, Z| = |Z, - 1 + 21
M ^ M , = | Z 2 - Z , | . D o d 6 : | Z , - 1 + zi = Z ^ - Z ,
Bai 4: Cho Z = a + b i . T i m tap hgp cac diem bieu dien ciia Z thoa dieu kien Z - 2 - i = 1
G i a i
Goi 1(2,1) la diem bieu dien ciia so phuc: 2 + i Z - 2 - i
I
( a - 2 + b - l ) i | = J( a - 2 ) ' + ( b - l ) ' = 1 • : ' < ^ ( a - 2 ) ^ + ( b - l ) ^ = 1
Do do tap hop cac diem M(a,b) ( Bieu dien so phuc Z) la d u o n g tron tam I ban kinh R=l
Bai 5: Cho A,B,C Ian l u g t la diem bieu dien ciia cac so phiic Z ^ = 3 - i , = - 2 + 3i va Z^ = - 1 - 2 i . Chiing m i n h rang diem bieu dien ciia so phiic tong Z ^ + Zp + Z^ la trong tam AABC
G i a i
Ta c6: Z ^ + Zg + Z^ = 3 - i + (-2 + 3i) + (-1 - zi) - 0 . Bieu dien hinh hgc ta c6: O A + OB + OC = 0 o O la trong tam A A B C . Vay diem bieu dien ciia so phuc tong Z ^ + Z„ + Zc trong mat phSng phuc goc O cung la trgng tam tam giac A B C
3. Bai tap t u l u y # n
B a i l : T i m tap hgp cac diem bieu dien ciia so phiic z thoa man cac h^ thuc sau: a) z + i > I z - i
b) I z - 1 + i I < 4 c) 2 < l z - l + 2 i | < 3
G i a i
z + i I > I z - i I =i> + (y + 1)2 > x2 + (y - 1)2 o y > 0
Tap hgp cac diem M la nua mat ph§ng phan 6 phia tren true Ox.
b) I z - 1 + i I < 4 (x - 1)2 + (y - 1)2 < 16; tap hgp cac diem M la hinh tron tam
1(1,1), ban k i n h R = 4 (tap hgp cac diem n i m trong d u o n g tron).
c) Tap hgp cac diem M(x, y) thoa man: 4 < (x -1)2 + (y + 2)2 < 9
. B a i l : Trong mat phSng (xOy) cho Ă-2, 0), B(0, 1), M va M ' Ian l u g t la diem z + 2
bieu dien ciia so phuc z va z' voi z' = z - i
a) T i m tap hp^p cac d i e m M sao cho O M ' = 1
b) T i m tap hgp cac d i e m M sao cho M ' 6 tren Ox
c) T i m tap h g p cac diem M sao cho M ' 6 tren Oỵ
Giai
a) O M ' = 1 z I = z + 2
z - 1
A M
B M = 1 <=> A M = B M . Tap hgp cac diem M la dirang trung true doan AB. b) Dat M(x, y) la diem bieu dien cua z
X + 2 + y i _ x^ + y^ + 2x - y + (x - 2y + 2)i
Ta CO z '
x + ( y - l ) i x^+iy-lf
M ' e Ox • » X - 2y + 2 = 0: tap hgp cac diem M la d u o n g thang y - 2y + 2 = 0
( d u o n g t h i n g qua ABO),
c) M ' € O y <=> x^ + y2 + 2x - y = 0: tap hgp cac diem M la d u o n g tron tam
- ) ban k i n h R = j -
Bai 3: Cho cac diem A, B, C, D Ian l u g t la diem bieu dien cua cac só phíic;
2-i, 3+2i, - l + 4 i va -2+ị C h u n g m i n h A B C D la h i n h b i n h hanh
Giai
A B = 1 + 3i, D C = 1 + 3i => A B C D la hinh b i n h hanh.
Bai 4: T i m tap hgp cac diem M bieu dien ciia sóphuc z thoa m a n dieu kien
a) z + 1 + 2i c) I z - 2 + i I = 1 c) I z - 2 + i I = 1 z + 2 b) z - 3 i = 2 Giai a) D u o n g t r u n g true doan A B v o i A ( - l , -2), B(-2, 0) b) D u o n g tron tam 1(0, 3), R = 2 c) D u o n g tron tam ]{-2, -1), R = 1
Bai 5: Cho z = x + y i (z i). T i m tap hgp cac diem bieu dien ciia z thoa man
z + 1 Z - 1 7 la so thuc d u o n g . , z + i x^ + - 1 + 2xi z - 1 x ' + ( y - i ) ' Giai só thuc d u o n g k h i x = 0
> 1 . Tap hgp cac diem M nSm tren tryc O y c6 t u n g d g thoa m a n y > 1
312
Cty TNHHMTVDVVHKhang
pai 6: T i m tap hgp cac diem bieu dien ciia so phiic z thoa man: 2 < | z- l + 2 i | < 3
Giai
Tap h g p cac diem M(x, y) thoa man 4 < (x - 1)^ + (y + 2)^ < 9 ' z + i
Bai 7: T i m tap hgp cac diem bieu dien ciia sóphuc z thoa man: = 1
, z - 3 i
Giai
Tap hgp cac d i e m M la d u o n g thSng y = 1.
y£n de 2: X A C D I N H P H A N T H l / C , P H A N A O , M O D U N
phtfong phap: Thuc hien cac phep tinh lien quan ciia bieu thuc de dua ve dang:
z = a + bi t u do suy ra phan thuc, phan ao ciia so phirc B i l l : T i m phan thuc, phan ao ciia cac so phiic sau:
a) ( l + 2 i ) ^ b ) ( l + i ) ^ + 2i c) (2 + i)(l - i)i d) Giai T + 2i 1 - i a) z = ( l +2i)2 = l - 4 + 4i = -3 + 4i Vay a = -3; b = 4 b) z = ( l + i p + 2i = l + 3i + 3i2 + i^ + 2i = -2 + 4i Vay a = - 2 , b = 4 c) z = (2 + i)(l - i ) i = (2 + i)(i + l ) = l + 3i V a y a = l , b = 3. d)z = _ l + 2 i _ ( l + 2 i ) ( l + i ) _ - l + 3 i _ 1 3 . 1 - i ~ 2 ~ 2 ~ 2 ^ 2 ^ Vay a = — , b = - 1 • y 2' 2
5 ^ : T i m phan ao cua sóphuc z cho biét: z = (72 + i)^(l - 72 i)
Giai
Taco: z = (^2 + i)^ (1 - ^ i ) = (1 + 2 V2 i ) 0 - V2i)
z = 5+ V 2 i= > z = 5 - N/ 2 i
Vay phan ao cua z la b = - \/2 ,
tCho sóphuc z thoa man: z = . T i m m o d u n cua so phirc: z + iz
Cdm nang Iuy4n thi DH - Nguyen ham - Tich phdn - So I'lnn Trdn Bd Ha
Giai
~ = (1 - >/3i)^ 1 - 3V3i + 9i^ - 3>/3i^ _ -8 _ -8(1 + i)
^ ' 1 - i ~ 1 - i 1 - i 2
z = -4 - 4i do do: z + iz = -4 - 4i + i(-4 + 4i)
Hay: z + iz = -8 - 8i I z + iz | =764 + 64 = 8 V2 . _
Bai 4: Cho so phuc z thoa man dieu kien: (2 - 3i)z + (4 + i) z = -(1 + Si)^. Tim phan thuc, phan ao cua sophuc z.
Giai
Dat z = X + yi
Ta c6: (2 - 3i)z + (4 + i) z = -(1 + 3i)2
(2 - 3i)(x + yi) + (4 + i)(x - yi) = -(1 + 3i)2
<=> 2x + 3y + (2y - 3x)i + 4x + y + (x - 4y)i = -(-8 + 6i) <=> 6x + 4y - (2x + 2y)i = 8 - 6i
6x + 4y = 8 2x + 2y = 6
Vay z = -2 + 5i nen phan thuc la -2 va phan ao la 5.
Bai 5: Cho so phuc z thoa man I z | = 1, z 5^ 1. Hay tim phan thuc, phan ao cua
<=> \ <r> X = -2, y = 5 só phuc z + 1 z - 1 Giai Dat z = a + bi, ta c6: a ^ + b ^ = l a + bi 1 z + l _ a + 1 + b i _ ( a + l + bi)(a - 1 - bi) b _ . z - 1 a - l + bi ( a - l ) ^ + b ^ 1-a z+1 ^ ^ —b Do do so phuc CO phan thuc bang 0 va phan ao bang
z - 1 1-a
^ , , , . , > . - u ' y3 + i
Bai 6: Tim phan thuc, phan ao va modun cua so phuc: z = + • 1 - i 2i Giai ^ _ ( N/ 5 - i ) ( l + i) ^ (V3 + i)(-i) _ 2 + N/5 ^ V s - V s - l . 1 - i ' -2í , „ 2 + V5 >/5-73-l Vay a = va b =
r jggi_7: Tim so phuc z c6 modun nho nhat thoa man: | z + 1 + 2i | =1 Cty TNHHMTl IHIH Khang Viel
Giai
Goi z = X + yi va M(x, y) la diem bieu dien so phuc z. Ta C O I z + 1 + 2i I = 1 <=> (x + 1 )2 + (y + 2)2 = 1
Duong tron (C): (x + 1)2 + (y + 2)2 = 1 c6 tam I(-l; -2) Duong OI CO phuong trinh y = 2x.
I So phuc z thoa man dieu kien tren va c6 modun nho nhat khi diem bieu dien cua no 6 tren (C) va gan goc O nhat, do la mgt trong hai giao diem cua OI