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Chuan kien thuc Hinh hoc 12 (Day them)

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W


!_J)'< -.9A)45,6D
"!&4&.9A
(V-.9A3)D01R9,)36"F %910"F
α
_)'< -> 
.9A


-



aa
%&4&.9A
X-.9A3)D01R9,)36"F #9C ,"E)36"FUW
W

!&4&.9A
"!&9+-5H"E,-)36
Y-+/019)361$% 53J,#1 #01

G1!#9C ,"E0%
)36"FUW
W

!5

G01!
"!&4&+/01
(W-9A3)D01R9,)36"F %9C 5H"E8A)365*9UW

W

!&4&.9A
"!&-c3C 1%5AG0R!
K

(-9A019)361$% 53)D, #,"E01

G1!#9C 5H"E
G0!%)36"FUW
W

!&4&.9A
"!j3)k)45Q3)D3)I> 9A%&Q1
(-9A01R9)361R$%?, #Q$%)45> 1#0Q

G1R!#9C 5H"EG0R!%)36"FUW
W
&4&.9A
((-9A 53h19" ,h1#h#h)?5*?9 %h1 #
h"#h&)'< -h> 9A
(K- 531?JZ1%1 E)'<Oe %?9G1!$b6
)45R[ --R HAOe ?9R#`R,g%`1R,f&4&
.+/Rfg
(S-9A 53)D019,)361 3,"E01#0#0,-)365*
9UW
-
R$% -)45> 015HAOe %?901
!&I[.4&>  .9A0R%01
"!&4&> .9A0R

(U-9A 530191S #U #1V 35H"E01#0#01,-
)365*9UW
-
&4&> .9A
(V-9A3)D01R#)36$%?, #,"E,-)365*9UW
-

$%)450HAO)e 1%[-[-R#`0,f%`0R,g&4
&.9A01fg
B. MẶT NÓN, MẶT TRỤ, MẶT CẦU.
I) MẶT NÓN, HÌNH NÓN, KHỐI NÓN:
1) Mặt nón:
- )'<O∆%+` ,h
%,-%9αGWlαlYW
W
!Hm
a- 6[ "Z)'<O+e 6
e )'<O∆$%5H9
* d: đường sinh
*

: trục
* O đỉnh
* 2
α
: góc ở đỉnh
2) Hình nón:
9ma- 6$%[ "Z5*
 53?e 6e 5*,
9?

nDiện tích xung quanh:0
ae

π
l
l: độ dài đường sinh
r: bán kính đường tròn đáy.
3) Khối nón:
9A-> 9
)'8$%.9
nThể tích khối nón: 
π
(




h: độ dài đường cao
S

r: bán kính đường tròn đáy
II) MẶT TRỤ, HÌNH TRỤ, KHỐI TRỤ:
1) Mặt trụ:
- )'<O∆%+[-[-
 %3 5*-c"F
Hma- 6["Z)'<O+
e 6e ∆$%5HN
* d: đường sinh
*


: trục
2) Hình trụ:
Nma- 6$%[ "Z5*
C@e 6e 5*,
nDiện tích xung quanh:0
ae

π
l
l: độ dài đường sinh
r: bán kính đường tròn đáy.
3) Khối trụ:
NA-> 9
)'8$%.N
nThể tích khối nón: 


h: độ dài đường cao
r: bán kính đường tròn đáy
 Chú ý: )..Nl.
III) MẶT CẦU, HÌNH CẦU, KHỐI CẦU:
1) Mặt cầu:
-)45h.)k%[.o
@A8A3)45-? 
3)45h5*-c"F)'8
$%5HJ5h"3&
p&/0Gh#!
{ }
h
=

Chú ý: nh1q

1F5-%G0!
nh1l

1F5-G0!
nh1

1F5EG0!
2) Vị trí tương đối của mặt phẳng và mặt cầu:
-5H0Gh#!%5HAOGr!$%> hE5AGr!%+h$%-c
3Th)5AGr!
n+q

Gr!?`G0! 6Gr!

G0!
φ
n+

Gr!AasG0!,
p)9(S): tiếp diện, (H): tiếp điểm
n+l

Gr!`G0!P-)'<mG!9J5#"3&

+

U


 Chú ý:+W 6h≡Gr!`G0!P-)'<mGh#!
3) Vị trí tương đối của đường thẳng và mặt cầu:
-5H0Gh#!%)'<O∆$%> hE∆%+h$%-c3
Th)∆
n+q

∆?`G0! 6∆

G0!
φ
n+

∆AasG0!,
p)9∆: tiếp tuyến#(H): tiếp điểm
n+l

Gr!`G0!, )45AJ"/1#
K!R/&ae #4&.
nR/&ae 0
ae
K
π



n4&.
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K
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BÀI TẬP
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)'8,-E
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 &+/&ae %4&> 9)9
K!*N9"3&)36%D -
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"!&4&> .N
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N"F(W
W
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S!`9)I0"Z5HAO)e N )'85* 53?J9,6D"F


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"!-+J6> )'<m)369[ --5HAOG0!,-5HAO
 )3695*9UW
W

&+/& 530
U!HAO)e N> N#`NP-+/$%?,_
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"!&4&> .N
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W
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V
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W!*.N$%*A-5*. )'<m)36> .NF5E5H> 
.
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")'< -> .N$%
"!&3k$b> 4&.N*A-."3&_-'
!-$@AA'B1R1222R2, 
!&+/&ae %4&> > .N9)'<m>  )36-,A3
?1R%1222R2
"!&+/&ae %4&> .99)I$%J5h> ?1R%
)36$%)'<m*A?1222R2
!-$@AA'B1R1222R2, j3)kJ5%"3&5H)e X
)I> $@AA'B)^-
(!-+/R19R1⊥G1!%R1S # 531?,%1( #K 
j3)kJ5%"3&5H)e ".)I> +/
K!-9A 53)D019,)36"F #,"E"F"
j3)kJ5%"3&5H)e 3)I9A

S! -+/R19R1⊥G1!%R1K # 531?,%1U #
X j3)kJ5%"3&5H)e ".)I1###R> +/
U!-9A01R9)361R$%?, #01⊥G1R!%01 #
j3)kJ5%"3&5H)e S)I0#1###R
V!-9A3)D01R9,)36"F #,"E"F"j3)kJ5%
"3&5H)e S)I0#1###R
X)-$MN112229bc3,)D"F 
! j3)kJ5%"3&5H)e 3)I> $MN
"! &+/&5H%4&> .'B
Y! -9A01R9)36$%?, #01⊥G1R!Ro5AGr!e 1%
?90HAOGr!`0#0#0R,2#2#R2
! _V)451###R#12#22#R2$?F5E5*5H
"! &+/&5H%4&> .)'8,-%
W! -9A 53)D019,)36"F #5H"E8A)365*9"FUW
W

! j3)kJ5%"3&5H)e 3)I> $MN
"! &+/&5H%4&> .'B
! -9A0190100 %9D -
! j3)kJ5%"3&5H)e 3)I> $MN
"! &+/&5H)9
X

PHƯƠNG PHÁP TỌA ĐỘ TRONG KHÔNG GIAN
QHỆ TỌA ĐỘ TRONG KHÔNG GIAN
A. KIẾN THỨC CẦN NHỚ
/ )*-? 
/t5" Nha#h6#hu)*5*?9)'8$%/N )*?9
-? 
] $b6" PB)Bk

→→→
kji ##
$$'8Eha#h6#hu
W#

======
→→→→→→→→→
ikkjjikji
 )*> )45%> PB
Gav6vu!
→→→
++=⇔ kzjyixOM 
→→→→→
++=⇔= kzjyixuzyxu !vvG
-1Ga

v6

vu

!#Ga

v6

vu

!
!vvG

zzyyxxAB

−−−=⇒
(PB"F  )*> PBd#/
-
!vvG!#vvG

zyxvzyxu ==
→→
n





=
=
=
⇔=
→→



zz
yy
xx
vu
n
!vvG

zzyyxxvu ±±±=±
→→

n
Rkkzkykxuk ∈=

!vvG

n
!#G!vvG

Rnmnzmznymynxmxvnum ∈+++=+
→→
K PBA'B

→→
vu#
A'BG









!W!G::
z
z
y
y
x

x
kzz
kyy
kxx
ukvRkuvu ==⇔





=
=
=
⇔=∈∃⇔≠
→→→→→→
S ) OP-I[.-'
 ) 1P-I[.











=



=


=
⇔=⇔≠
k
kzz
z
k
kyy
y
k
kxx
x
MBkMA
BA
M
BA
M
BA
M




$%)451







+++

v

v

BABABA
zzyyxx
U&?'>  PB
- PB
!vvG!#vvG

zyxvzyxu ==
→→
Y



#-[wwww zzyyxxvuvuvu
++=






=

→→→→→→
w







w zyxuu ++==
→→
1

!G!G!G
ABABAB
zzyyxxAB
−+−+−=


!Www#WwGw


wwww

!#-[G














→→→→
→→
→→
→→
≠≠
++++
++
==
vv
zyxzyx
zzyyxx
vu
vu
vu

W

=++⇔⊥
→→
zzyyxxvu
V&9'>  PB
x









=
→→






vvy#
yx
yx
xz
xz
zy
zy
vu

→→→→→→
⊥⊥ vvuuvu y#x#y#x

!#[Gwwwwwy#xw
→→→→→→

= vuvuvu

→→
vu#
A'B
→→→
=⇔
Wy#x vu

→→→
wvu ##
)tAO
Wy#x =⇔
→→→
wvu
X3+N

[ ]
ACABS
ABC
#


=

[ ]
\#
\\\\
AAADABV
DCBAABCD

=

[ ]
ADACABV
ABCD
#
U

=
YH
zH3J5QG v"v!9"3&_9A'B
Ga{ !

|G6{"!

|Gu{!

_

z]'8$,#A'Ba

|6

|u

| a|"6|u|+W$%A'B> 5H
9)D/

|"


|

q+p)9QGz vz"vz!$%J5> 5H%"3
&_
dcba
−++

B.BÀI TÂP.
:-
!KvWvKG!#vvG!#(vvG −=−==
→→→
wvu
5 )*

x
#"
!
→→→→→→→→→→→→→
=+−+−−=−+=
W(!#


(S!#K xwvucwvuxbwvux
:-

u
9)45)$%Gvzv(!%)45.$%Gzv(vS!
-3PB[ )J6PB%-A'B

u


W

→→→→→→→→→→→
+−=+=++−= kjicckjbbkjiaa K!#K!#KXU!
(:-" )451GvSv(!#G(vVvK!#Gav6vU!5a#6)41##O%
K:- )451GzvUvU!#G(vzUvz!5*5AGha6![ --1|}b
S:5".)451Gvzv!#Gv(v!#GKv(v!#RGKvzv!$%3)I> C@
&)*+%3)'<=-#a3)kJ5> C@)9&-[> 9C  PB
# BDAC
U:&&9'
y#x
→→
vu
"
→→→→→→→→→→
+−−=−+=−=−= kjivkjiubvua (#(!!vvKG#!(vvG!
→→→→→→→→
−=+=−=−= jivkiubvuc #K!!KvWv(G!#vvWG!
V:&
→→→
wvu y#x
"
!vvG!#(vvKG!#v(vWG! −=−−==
→→→
wvua
→→→→→→→→→→→
+−=+=−+= kjiwkjvkjiub (#S#(K!

→→→→→→→→→

=++=+= iwkjivjiuc ##!
X:}".)45[ )J6$%".)> "%%&+/&> "%)9
1Gvv!#Gv(vK!#GUvSv!#RGVvVvS!
Y:5Eh6)453)D )451G(vvW!%GzvKv!
W!:5E5HAOhau3)D" )451Gvv!#GzvvW!#G(vvz!
:- )451GvzvV!#GKvSvz!;'<O1`5AGh6u!,)45;45 ) 
1P-I[.%-~5 )*)45
:j=[o)tAO> " PB
→→→
wvu ##
-5i'<8A[ 
!
!(vvKG!#vvWG!#vvG ==−=
→→→
wvu
"!
→→→→→→→→→→→
+=++=++= kiwkjivkjiu #((#SK
(:-" PB
!Kvv(G!#v(vG!#WvVv(G −===
→→→
wvu
! 5
→→→
wvu ##
?O%
"! 4k
!(vvKG −−=

a

P-" PB
→→→
wvu vv

K: !-
!(vvG!#vvG =−=
→→
bma
55)4
→→

ba
"!-
!Kv($-v(G!#vS$-vG
S(
==
→→
bma
55)4

→→

ba
!-
!WvvG −=

a
5

b

A'B

a
#"F
W
=
→→
ba

S:-?  )*ha6u-".)451GvvW!#Wvv!#GvWv!#
RGvv!
! 5".)45?)tAO&4&+/1R
"! 5 )*J5>  531#J5+/1R
! &+/&35O> +/


+! &)*+%3)'< -> .+/
P! &9C  )'<O1%R
•! A'B5H-,A+/1R
U:-?  )*ha6u-" )451GvWvW!#GWvWv!#Gvv!
! 51##$%" )I> 5* 53
"! &%+/&>  531
! 5 )*)45R)41R$%"%
+! &)*+%)'< -

>  531
P! &39>  531
•! j3)k )*oJ5>  531
! j3)k )*J5)'<m A 531
V:-".)451GvzvU!#Gz(vzvzK!#GSvzvW!#RGvv!

! 51$% 53?
"! &"3&)'<m*# A 531
! &)*+%)'<AJ3->  531€T)I
X:-+/1R9)I1Gvvz!#G(vWv!#Gvzv(!%R*Nh6
1R
S
& )*)IR
Y:-$@AA'B1R12222, 
! 512
!\\G DAB


"! $%)451R#]$%)452512
MN


! &-[> 9C  PB
MN
%
\AC

+! &
12]

W:A'B5H-3'<8A[ 
! J5QGvWvz!#)'<&"FX
"! ;'<&11Gzvv!#GWvv(!
! J5hGWvWvW!Aas5HJ5QG(vzvK!%"3&_
+! J5QGvzv(!%)e 1GVvv!
P! J5QGzvv{(!%Aas5AGha6!

•! J5QGzvvz(!%AasNh6
! J5QGzvvz(!%"3$%hQ
:-3A'B[ A'B%-$%A'B> 5H
! a

|6

|u

za{U6{Xu|W
"! a

|6

|u

{6W
! a

|6

|u

{a{K6|UuzW
+! a

|6

|u


{(a|K6{Xu|SW
!A'B5H-3'<8A[ 
 !;e " )451GvvzK!#Gvz(v!#Gvv(!%9J5F5E5AGha6!
"!;e  )451G(vzv!#Gvvz!%9J5*Nhu
!;e ".)451Gvv!#Gvv!#Gvv!#RGvv!
(:-A'Ba

|6

|u

{K5a|K6|5u|5

|K5W55)49$%A'B5*5H
%55)4"3&5H$%}b


II.PHƯƠNG TRÌNH MẶT PHẲNG
QKIẾN THỨC CẦN NHỚ.
PBA3A6> 5HAO
nPB
→→
≠ Wn
)'8$%r> 5AG
!
α
9F5E)'<O?9
5AG
!
α

#`$%
!G
α


n
n]
!vvG!#vvG

zyxvzyxu ==
→→
?A'B%3)'<O 
s[-[-G-HF5E!5*5AG
!
α
G
→→
vu#
m$%HAPBIA'B
> 5AG
!
α
!









=






=
→→→






vv#
yx
yx
xz
xz
zy
zy
vun
$%5*r> 5AG
!
α

r'Bde3
1a|6|u|RW1


|

|


W

r
G
=

n
1vv!
(5A
W!G!G!G!G
!vvG
!vvG
!G
WWW
WWWW
=−+−+−⇒





=

zzCyyBxxAmp

CBAnVTPT
zyxMqua
αα
K'<8A)H"/-5AG
!
α
1a|6|u|RWp)9
nRW

G
!
α
)e . )*
nW#R
!GW
α
⇔≠
[-[-Nhu
RW
!G
α

 Nhu
nW#R
!GW
α
⇔≠
[-[-5AGh6u!
nRW
!G

α

&$%5AGh6u!
G3'<8A3[6 'Bo!
Sk&'B).>  5HAO
- 5HAO
!G
α
1a|6|u|RW%
!\G
α
12a|26|2u|R2W
\\\\
!\::G!G
D
D
C
C
B
B
A
A
≠==⇔
αα
\\\\
!\G!G
D
D
C
C

B
B
A
A
===⇔≡
αα
\\\\\\
!\G!G
A
A
C
C
hay
C
C
B
B
hay
B
B
A
A
≠≠≠⇔∩
αα
W\\\!\G!G
=++⇔⊥
CCBBAA
αα
• s7 e6'5*•AJ[.‚%-)99•5ƒ[.•"FW•„[.
•…"FW

Ur'BP-) `> 5HAO
AG
!
α
`ha,1G vWvW!#`h6,GWv"vW!#`hu,GWvWv!9A'B
$%
(

W# ≠=++ abc
c
z
b
y
a
x
V9>  5HAO
- 5HAO
!G
α
1a|6|u|RW%
!\G
α
12a|26|2u|RW

ϕ
$%9>  5HAO# 9

\\\
\\\
-[

CBACBA
CCBBAA
++++
++
=
ϕ
Xp} 3T5*)45)5*5HAO
-5AG
!
α
1a|6|u|RW%)45
W
Ga
W
v6
W
vu
W
!p)9
+G
W
#G
!
α
!

WWW
CBA
DCzByAx
++

+++
QQBÀI TẬP.
-5i'<8A[ A'B5HAOGr!
!† )45
W
Ga
W
v6
W
vu
W
!%$$'8[-[-35HAO )*Gha6!#Gh6u!#Ghau!
"!† 3> )45
W
Ga
W
v6
W
vu
W
!Ga
W
6
W
u
W

W

!#$E3ha#h6#hu

!† )45
W
Ga
W
v6
W
vu
W
!Ga
W
6
W
u
W

!W

%$$'8 3Nhavh6vhu
-5i'<8A[ #A'B5HAOGr!
!;e " )451Gzvv(!#GvzKv(!#GKvSvU!
"!;e )45
W
Gv(vz!%?9Nh6
!;e 
W
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+!;e Gv(v!%[-[-5HAOa{6|u|KW
P!;e  )451G(vvz!#GvzvK!%?95HAO
a{6|uW
!;e 

W
Gvzv!#[-[-Nhu%?95HAO
a{6|(u|W
!;e 
W
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(5 )4".)451Gvv!#Gv vW!#Gvzv!#RGvv!*5*5HAO
K-" )451Gvv!#G(vzv!#GzvWv!;459*5HAOo> ) 1
?~
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U- 5HAO9A'Ba{56|(u{UW%G5|(!a{6|GS5|!u{WW
3k%-> 5 5HAO)9
z 0-[- 
z  
z ` 
z ?9 
K
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VA'B5HG0!9J5QGzvv!%Aas5HAO9A'B
a|6{u|SW
X-)45".)451G(vzvz!#G(vvW!#GWvv!#RGzvv!A'B5HJ51#
Aas5AGR!
YA'B5H)e " )451GvWvW!#GWvvW!#GWvWv!%9J5QF5E5H
AOa|6|u{(W
W-5HG0!a


|6

|u

zUa{6|Ku|SW%)45GKv(vW!A'B5HAO
Aas5H,
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-*AC@1R1222R291GWvWvW!#G vWvW!#RGWv vW!#
12GWvWv"! #"$%C[.+'B%$%)452
! &4&+/R12
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b
a
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III.PHƯƠNG TRÌNH ĐƯỜNG THẲNG
QKIẾN THỨC CẦN NHỚ.
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+





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!vvG
WWWW

cbauVTCP
zyxMqua
9
zr'B 5[.> +
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W
W
Rt
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WWW


=

=


abc
c
zz
b
yy
a
xx
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W
W
uVTCP
Mqua
d
uVTCP

Mqua
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Wy\#x
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WW
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→→→
→→
Wy\vx
Wy\x
\
WW
uu
MMuu
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=

→→
→→→
Wy#x
Wy\#x
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WW
MMu
uu
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→→→→
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Wy#xy\#x\
\
WW
MMuuud
S

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Wy\#x
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WW
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→→
MMuu
(k&'B).> )'<O5HAO
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!vvG
WWWW
cbauVTCP
zyxMqua
%5AG
!
α
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W!G
≠++⇔∩
CcBbAad
α
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W
W
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WWW
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d
α
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⇔⊂
W
W
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WWW
DCzByAx
CcBbAa
d
α
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→→→→→
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α
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!vvG cbau =


%+29r
!\v\v\G\ cbau =


9
ϕ
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ccbbaa
uu
uu
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α
9r
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
ϕ
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!
α
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cbaCBA
CcBbAa
nu
nu
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++
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Ga

v6

vu

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uVTCP
Mqua
W
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!
α
e 

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α

z +G

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U
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\#
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W
W
uVTCP
Mqua
uVTCP
Mqua

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!

α
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\
W
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!
α

z
!!G#G!\#G
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W
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|Cách 2













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→→
→→
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\
WW
uu
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d
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→→→→
++−= kjiu S(
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−=
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tz
ty
tx
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ty
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!




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W!G#

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W
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W
GKvz(v!E)'<O
X
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+
z
yx
−=
+
=
+


(

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W
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W
Gvzv!e )'<O





=
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ty
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tz
ty
tx
d
tz
ty
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BÀI TẬP TỔNG HỢP
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:&9C  5HAO%A'B 5[.>  -6>  5HAOGr!
%G†!

:A'B5HAOG_!)e . )*h?9Gr!%G†!
Y:-? / )*ha6u#-)45RGz(vv!%5HAOGr!)e " )45
1GvWv!#GWvvW!#GvvX!
:A'B)'<O1%A'B5HAOGr!
:A'B5HJ5R#"3&_S5F5H%6`5HAO
Gr!
W:-? / )*ha6u#-5HAOGr!a|6|u|SW%5H
G0!a

|6

|u

{a{K6|KuW
:5J5%"3&> 5HG0!
:A'B5HAOG†![-[-Gr!%AasG0!5 )*> A
)45
:-? / )*ha6u#-" )451GvKvW!#GWvv!#
GvWvzK!
:5 )*)45R)41R$%"%%5 )*J5> "%
:A'B)'<OG+!)e J5>  531%?9
5AG1!
:-? / )*ha6u#- )'<O+

(







=

=

− zyx
#


+2





−−=
−−=
=
tz
ty
tx
(
S
:5+%+2=- 
:A'B5HAOGr! +%[-[-+2&} 3C +%+2
(:-? ha6u#-".)451Gvzv!#GvWvW!#GWvvW!#RGWvWv(!
:A'B5HAOGR!06 1R$%5*+/
:5)4512[ --5AGR!$%5HAOo> ) 112
K:-? / )*ha6u#-)'<O+






+
=

=
zyx
% 5HAO
Gr

!a|6{u|SW#Gr

!a{6|u|W
:&9C 5AGr

!%5AGr

!#9C )'<O+%5AGr

!
:A'B5HJ5Q*+%Aas5AGr

!%5AGr

!
S:-? / )*ha6u#- )451Gvv!#GvzvS!
:A'B5HG0!)'<&1
:5)45E)'<O1[ -- 53h1?,h

U:-? / )*ha6u#-5HG0!a

|6

|u

{a{K6{UuW% )45
Gvv!#]GvzvS!
:5J5Q%"3&_> 5HG0!A'B5HAOGr!e 3
> J5QE3N )*
:})'<O]`5HG0!, )455 )*3 -)45)9
V:-? / )*ha6u#- )451G(vWvz!#GvzvK!
:A'B)'<O1%A'B5HAOo> ) 1
:A'B5HJ51%)e )455)45).a> e 1
X:-? / )*ha6u#- )'<O+
K

(




=
+
=

zyx
%+2






+=
+=
+−=
tz
ty
tx
KK
(


:5+[-[-+2&} 3C +%+2
:A'B5HAOGr! +%+2
Y:-? / )*ha6u#-)451Gvzv(!#5HAOGr!az6zu|W%
)'<O+
(


 zyx
=


=


:5 )*)4512).a> 1e 5AGr!
:5 )*> )45E)'<O+[ --} 3T)5AGr!"F(
W:-? / )*ha6u#-)451Gvv!#5AGr!a|6{u{W%)'<O

+






==

zyx

:5)4512).a> 1e +
:A'B)'<O)e 1#[-[-5AGr!%`+
:-? / )*ha6u#-)'<O+



K

+
=

=
zyx
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