các bai hinh hoc giai tich thi dai hoc

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
( )
   C CD x y C t t∈ + − = ⇒ −
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
 

 
t t
M
+ −
 
 ÷
 

( )
 
        
 
t t
M BM x y t C
+ −
 
∈ + + = ⇒ + + = ⇔ = − ⇒ −
 ÷
 
 !"#$%
  AK CD x y⊥ + − =
&'!

K BC
∈
"

( ) ( )
     AK x y x y− − − = ⇔ − + =
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()'*+*,
( )
 

 
x y
I
x y
+ − =

⇒

− + =


-./&0'.
⇒
()
( )
K −

12*345#.067*189*


:  : 
  
x y
x y
+
= ⇔ + + =
− +

 !"#$%  
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3$4$
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( )
 ;AB AB= − ⇒ =
uuur
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9*4
  x y+ − =
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( ) ( )
 I d y x I t t∈ = ⇒
'=
4> 06
( ) ( )
  #   C t t D t t− −

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>
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ABC

S AB CH= =
!@*AB"
:
;
CH⇒ =
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CB
( )
( ) ( )
: ;   
 # 
D E : D :
    

; ;
  #  
t C D
t
d C AB CH
t C D

   
= ⇒
−
 ÷  ÷

= ⇔ = ⇔
   



= ⇒ − −

FG()=>
;   
 # 
   
C D
   
 ÷  ÷
   
*B?
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 #  C D− −
/)#56+7!18.,
∆
!2!&')99,:;
  x y+ + =
6<=);
 x y+ − =
>6&?)!&'+
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7JKL 
 #=8JKMN
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7
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x y
a b
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  
    ab
a b ab
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I
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x y
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Câu VIa. (2.0 điểm)
/)#56+7!18., !&'+

;.O@,O-

;.P@,O-

/#7!1<#=Q!&')R16 #=@7#)*


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)G
8,
3$4$
R(BK

=K

6!"
R(4BK

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R(BK

=SU6!:"
R(4'127*/-B64=S'*)U$*66
'!:W"#XY:W
Câu VIb. (2.0 điểm)
ST6U;.
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PV,
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2LYQZ=[:\*9!"#W

\!&'+.-

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=)]QI!^
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

x =
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x−
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MF
MH
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( )
 
  C x y x+ + =
>
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( )
C
_ 69,,)G9
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( )

∆
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     C x y I R+ + = ⇒ − =
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>B6
( )

  x y b∆ − + =
^7Nd!"
( )

#d I R⇔ ∆ =


  

b
b
−
⇔ = ⇔ = ± +
.e
( )
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    x y∆ − ± + =


F
( )

  x y b∆ + + =
^7Nd!"
( )

#d I R⇔ ∆ =


  

b
b
−
⇔ = ⇔ = ± +
.e
( )

    x y∆ + ± + =

<9> ;1,0 điểm /)#56+0 !1Oxy !&'+
( )
  : d x y− − =
AB6
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3
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/12fc9
6
:
 : :#

m m m m= − ⇔ = =

.*
:

m =
*9<
 
: : E
  g
x y
   
− + + =
 ÷  ÷
   

.*
:m
=
*9<
( ) ( )
 
: : Ex y− + − =


<9> ;1,0 điểm/)#56+0 !1OxyAB66&?)!&'+C9
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 
=
 ÷
 
uur
-
=h87*-7^#KB6-667*189*c1k
•

!N " ! "  N : : 
:
, hay : − − + − = − + =

∆
!2!&')99,:;
  x y+ + =
6<=
);
 x y+ − =

 >6&?)!&'+
3$4$

( )
   C CD x y C t t∈ + − = ⇒ −


 

 
t t
M
+ −
 
 ÷
 


( )
 
        
 
t t
M BM x y t C
+ −
 
∈ + + = ⇒ + + = ⇔ = − ⇒ −
 ÷
 
 !"#$%

  AK CD x y⊥ + − =
&'!
K BC
∈
"

( ) ( )
     AK x y x y− − − = ⇔ − + =

()'*+*,
( )
 

 
x y
I
x y
+ − =

⇒

− + =


-./&0'.
⇒
()
( )
K −


12*345#.067*189*

:  : 
  
x y
x y
+
= ⇔ + + =
− +
 !"#$%  !&'(#
)*!&'+,-./#0 !1!2
3$4$
6
( )
 ;AB AB= − ⇒ =
uuur
<*189*
4
  x y+ − =

( ) ( )
 I d y x I t t∈ = ⇒
'=
4>06
( ) ( )
  #   C t t D t t− −

?$*-
>
 :

ABC
S AB CH= =
!@*AB"
:
;
CH⇒ =

CB
( )
( ) ( )
: ;   
 # 
D E : D :
    

; ;
  #  
t C D
t
d C AB CH
t C D

   
= ⇒
−
 ÷  ÷

= ⇔ = ⇔
   



= ⇒ − −

FG()=>
;   
 # 
   
C D
   
 ÷  ÷
   
*B?
( ) ( )
 #  C D− −
" /)#56+0 !1
Oxy
!&'+
! "d
6&?);
x y− =

!"#
!"M
/#6&?)!&'+
∆
D)G7
A
D!&'
+
! "d

7
B
L  #=
AMB
9I<7
M
3$4$
A
n0
Ox
0
( )
A a
#
B
n012*3
x y− =
0
!  "B b b
#
!"M
!  "# !  "MA a MB b b⇒ = − − = − −
uuur uuur
-4=`/&0
  
! "! " ! " 
 
! "  ! " ! "
a b b
MA MB

MA MB
a b b
− − − − =


=
 
⇔
 
=
− + = − + −




uuur uuur
#
KB
b =
$*`*+r=G

  
 

 # 

 # 




! "  ! " ! "
 ! " ! "

b
a b
b
a b
b
b
b
a b b
b b
b
−

− = ≠
−


− = ≠
−
 
⇔
−
 
−
 
 
− + = − + −
+ = − + −


 ÷

−
 

 



 # 



:
! " ! "   
! "

a
b
a b
b
b
a
b b
b
b

=


−

− = ≠



=
−
 

⇔ ⇔


 
=


 
− + − − =

 
 


−
  =





FS


a
b
=


=

12*3
∆
5467*189*
 x y+ − =
FS
:

a
b
=


=

12*3
∆
5467*189*
  x y+ − =
<9>$ (2,0 điểm)
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:/#7!1=!2% _B
3$4$
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
  
 
;

 :  
; ;
;
x
x y
B
x y
y

=

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
 
⇔ ⇒
 
 ÷
− + =
 



=


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! " ! " !  "
AB BD AC
n n n a b− −
uuur uuur uuur
!với a
2
+ b
2
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.*66
( ) ( )
B[ # B[ #
AB BD AC AB
c n n c n n=
uuur uuur uuur uuur
   

   


a b
a b a b a ab b
b
a
= −



⇔ − = + ⇔ + + = ⇔

= −

VFSYVL*(Y
⇒
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  
!"
   
x y x
A
x y y
− − = =
 
⇒ ⇒
 
− + = =
 
R('/*9**s*G*9'Y∩4>0B&)'*,*,

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 ;


 :  ;
 


x
x y
I
x y
y

=

− − =


 
⇔ ⇒
 
 ÷
− + =
 


=


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( )
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:  
; ;
C D
 

 ÷
 
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
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!  " ; !  "  
B B B B C C C C
B x y d x y C x y d x y∈ ⇒ = − − ∈ ⇒ = − +
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 E
 
B C
B C
x x
y y
+ + =


+ + =

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6
!:" !: "
BG
BG VTPT n⇒ −
uuur uuur
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4-$i*XYK!4R"Y
g
;

⇒
7*189*12f!No;"

q!o"

Y

;
<9> ;(1,0 điểm) /)#56+0 !1(Oxy)   #=ABC 9I<7A\
( )
A

( )
 G ;
T)0<#/=Q!&')R16 #=ABC.
GIẢI
?4Y
( )

 

 
 
ABC
a
a
BC a S p
+
→ = → = =

( )
     AG AG AM a= − → = → = → =
uuur
( )
  r→ = −


 
ABC
S
a
r
p
→ = =
+

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( )
 A ;

6&?) !&')99,%  #=ABCC9  !2B  CTjT&dT

  x y
− + + =

  x y
+ − =
./#0 !1 !"#B  C
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 
: 

 
 
x y
G
x y
− =

 
↔

 ÷
+ =
 


.e
   
  
   

B C
   
− −
 ÷  ÷
   

R(
( )

  ! "B b b d
− ∈

( )

   ! "C c c d
− ∈

6
; 

 
 

 
b c b
b c c
 
− = =
 
 

↔
 
 
+ = = −
 
 

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 
! "   o   o    #C x y x y+ + =
 
! t"   : o ; C x y x+ + =
 k!C9 M>6&?
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! "# ! t"C C
TjT&d7A, B L MA= 2MB.
GIẢI
qR(/=L-$i*!C"#!C"c1kI!"#I!V"=
# t R R= =
#12*3!d"
5M 67*189*

! " ! " # ! "!H"a x b y ax by a a b + = + = +

qR(H, Hc1kAM, BM.
.*66

t t tMA MB IA IH I A I H= =
( ) ( )

! " :ug ! t " vd I d d I d =
#
IA IH
>
( ) ( )



g
: ! t " ! " ; : ;
a b
d I d d I d
a b a b
= =
+ +



E
; E
a b
a b
a b

= =
+
>w*x
b
0*(
E

E
=

=

=

a
b
a

.A$,
IA IH>
y*=B!H"6*12*3*BZr
2)Trong không gian với hệ tọa độ Oxyz cho hai đờng thẳng d và d lần lợt có phơng trình :
d :
z
y
x
=


=


và d :

;

+
==

z
y
x
.
Viết phơng trình mặt phẳng
"!

đi qua d và tạo với d một góc


GII
.Đờng thẳng d đi qua điểm
"!M
và có vectơ chỉ phơng
"!

u
Đờng thẳng d đi qua điểm
";!t

M
và có vectơ chỉ phơng
"!t

u
.
Mp
"!

phải đi qua điểm M và có vectơ pháp tuyến
n
vuông góc với
u
và


EB["tB[!

==
un
. Bởi
vậy nếu đặt
"! CBAn
=
thì ta phải có :





=
++
+
=+

E



CBA
CBA
CBA






=
+=






+++=
+=

"!E


CACA

CAB
CCAAA
CAB
Ta có
""!!

=+=
CACACACA
. Vậy
CA
=
hoặc
CA
=

.
Nếu
CA
=
,ta có thể chọn A=C=1, khi đó

=
B
, tức là
"!
=
n
và
"!

mp
có phơng trình
"!
=++
zyx
hay
:
=++
zyx
Nếu
CA
=

ta có thể chọn
#
==
CA
, khi đó

=
B
, tức là
"!
=
n
và
"!

mp
có phơng

trình
"!
=
zyx
hay

=+
zyx
/)#56+\0 !1Oxyl,TB66&?)69,9% ST6
E;


E
x y
+ =
6 ) TP;y

-x
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qB

l"
!"^7^!E"A

qEB

YC

!"
!"^7^!P"B

Y:ACB

YAC!"
*^!"=B!"6CY:A*B?CYA
FSCYAAYBY!B&"
FSCY:A⇒


A
B = ±

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  
:  : 


A
Ax y A x y± + = ⇔ ± + =
FG6*^7^c9
 
: 

x y± + =
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( )
 


 : ; C x y y+ − − =

( )
 

 E  E C x y x y+ − + + =
AB66&?)69,9% 
( )

C

( )

C
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( ) ( ) ( ) ( )
     
  #    : # C I R C I R= − =
R(^7^*
( ) ( )
 
#C C

( )
 
  Ax By C A B∆ + + = + ≠
∆
^7^*
( ) ( )
 

#C C

( )
( )
( )
( )
 
 
 
 
  


 :  
B C A B
d I R
d I R
A B C A B


+ = +
∆ =
 
⇔ ⇔
 
∆ =
 
− + = +



 !"=!"[
A B=
*B?
 

A B
C
− +
=
Trường hợp 1:
A B=

*(
    ;     ; B A C x y= ⇒ = ⇒ = − ± ⇒ ∆ + − ± =
Trường hợp 2:
 

A B
C
− +
=
*=B!"1k
 
:
       :  g 

A B A B A A B y x y− = + ⇔ = = − ⇒ ∆ + = ∆ − − =
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
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