00_dang tang hs muc tieu 8-9-10_p2.pdf
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Kha hc Chinh phc Hnh phng Oxy v Kthut gii Hphng trnh Thy ng Vit Hng Moon.vn
Tham gia Luyn thi mn TONti MOON.VNt im scao nht trong k thi THPT Quc gia 2016
CHINH PHC IM 8-9-10 MN TON Phn 2Thy ng Vit Hng Moon.vn
VIDEO BI GING v LI GII CHI TIT CC BI TP chc ti website MOON.VN
PHN 1 : BI
Cu 1: Gii phng trnh ( ) ( ) ( )22 2 231 2 2 1 2x x x x x x = + + + .
Cu 2. Trong mt phng ta Oxy, chotam gicABCvung tiBcBC= 2AB. im M(2; -2) l trung
im ca cnhAC. GiN l im trn cnhBCsao choBC= 4BN. im4 8
;5 5
H
l giao imANv
BM. Xc nh ta cc nh ca tam gicABCbitNthuc ng thngx+ 2y 6 = 0.
Cu 3: Gii hphng trnh2 23
1 2 1
12 7 8 15
x y x y y
xy x x y
+ + + = +
+ = + +
Cu 4. Trong mt phng ta Oxy,chohnh chnhtABCD cAB= 2BC. GiHl hnh chiu caA
ln ng thngBD.E, Fln lt l trung im on CDvBH. BitA(1;1), phng trnh ng thng
EF l 3x y 10 = 0 v imEc tung m. Tm ta cc nhB, C, D.
Cu 5. Gii hphng trnh( ) ( )
2
5 3 2 2 3 4
2 2
x y x y x y x y x
y x
+ + = + + +
+ =
Cu 6. Trong mt phng vi htoOxycho tam gicABCvung cn tiAvi Ml trung im caAB,ng thng quaAvung gc viMCctBCtiH, bit phng trnh ng thng : 1 0AB x y + =
v trung im caHBl14
5;3
K
. Tm tocc nh ca tam gicABC bitBc hong ln hn 4.
Cu 7: Gii hphng trnh
2 2 2
33
2 4 2 3 4 2
3 2 12 1 3 2 2 1
2
x y x y y
x yx y x y
+ + + + = + + + +
+ = +
Cu 8. Trong mt phng vi h ta cho tam gic ABC vung ti B c phn gic trong AD vi15 1
;2 2
D
thuc BC .Gi E, F l 2 im ln lt thuc cc cnh AB v AC sao cho AE AF= . ng
thng EF ct BC ti K. Bit im 11 3;2 2
F
, E c tung dng v phng trnh ng thng
: 2 1 0AK x y + = . Tm tocc nh ca tam gicABC.
Cu 9: Gii hphng trnh
( ) ( )
2 2 2 2
2 2
2 33
5 6 5 6 8 6
2 4
x x y
x xy y x xy y
x y x x y
++ = + +
+ + =
PHN 2: LI GII CHI TITCu 1: Gii phng trnh ( ) ( ) ( )
22 2 231 2 2 1 2x x x x x x = + + + .
Li gii.
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Kha hc Chinh phc Hnh phng Oxy v Kthut gii Hphng trnh Thy ng Vit Hng Moon.vn
Tham gia Luyn thi mn TONti MOON.VNt im scao nht trong k thi THPT Quc gia 2016
iu kin 1 1x .
Phng trnh tng ng ( )22 2 232 2 2 2 1 1x x x x x x + = + .
Ta c ( )2
2 2 21 1 2 1 1 1 1x x x x x x + = + + .
t 3 2 2 2x x t = thu c ( )3 2 20
1 1 1 01
tt t t t
t
= +
{ }20 2 2 0 1 3;1 3t x x x= = + .
( )( )3 23
1 1 2 3 0 3 1 0 11
xt t x x x x x
x
+ =
.
i chiu iu kin ta c nghim duy nht 1x = .
Cu 2. Trong mt phng ta Oxy, chotam gicABCvung tiBcBC= 2AB. im M(2; -2) l trung
im ca cnhAC. GiN l im trn cnhBCsao choBC= 4BN. im4 8
;5 5
H
l giao imANv
BM. Xc nh ta cc nh ca tam gicABCbitNthuc ng thngx+ 2y 6 = 0.
Li gii:
GiEl trung im caBCv F AN ME = .
Ta c:3 3
22 2
HM MFEF AB ME HM HB
HB AB= = = = =
.
( )
6 3 4
5 2 50;4
18 3 85 2 5
B
B
x
B
y
=
=
.
Gi ( )6 2 ;N t t ( )12 4 ;2 4E t t . Li c: . 0EM EB=
( )( ) ( )( )
( )4;0212 4 10 4 2 8 2 2 0 17 8 14
;5 5 5
Et
t t t t t E
= + = =
.
Vi ( ) ( ) ( )4;0 8; 4 4;0E C A .
Vi8 14 16 8 36 28
; ; ;5 5 5 5 5 5
E C A
.
Cu 3: Gii hphng trnh2 23
1 2 1
12 7 8 15
x y x y y
xy x x y
+ + + = +
+ = + +
Li gii.
iu kin2
1 0; 0 0
2 1 0; 7 0
x y x y x
x yy xy x
+ +
+ +
Phng trnh thnht ca htng ng vi
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1 2 1 0 01 2 1
. 0 1 01 2 1 1 2 1
x yx y y x y x y
x y y
x y x yx y x y x y
x y y x y y
+ + + + = + =
+ + + +
+ = + = + + + + + + + +
Ta c 1 0 01 2 1
x yx y x y
x y y
+ > = =
+ + + +
.
Phng trnh thhai ca htng ng 3 3 212 7 8 15x x x x+ = + + .p dng bt ng thc Cauchy cho 3 sthc khng m ta c
( ) ( )2 2
32 2 3 23 38 8 7 8 15
4 7 8.8 7 12 7 8 153 3
x x x xx x x x x x x x
+ + + + ++ = + = + + + .
Do phng trnh n x c nghim khi 28 7 8 1x x x= + = = .Kt lun hc nghim duy nht 1x y= = .
Cu 4. Trong mt phng ta Oxy,chohnh chnhtABCD cAB= 2BC. GiHl hnh chiu caA
ln ng thngBD.E, Fln lt l trung im on CDvBH. BitA(1;1), phng trnh ng thng
EF l 3x y 10 = 0 v imEc tung m. Tm ta cc nhB, C, D.
Li gii:
Gi Kl trung im caABkhi AKEDl hnh
vung, giIl tm hnh vung.
Khi / /KF AH KF DF do vy 5 im
A,K,F,E,Dcng thuc ng trn ng knh KD.
Suy ra : 3 4 0AF EF AF x y + = .
17 1 32;
5 5 5F AF
=
Ta c:1 2
22 5
AFE DCB EF AF = = .
Gi ( )
( )
( )
2 22
3 3; 18 17 51 8
;3 10 3 19 19 75 5 5 5 ;
5 5 5
t E
E t t EF t tt E loai
=
= + = =
Khi : ( ): 2 0; 2;0 : 2 0AE x y I KD x y+ = = . Gi ( ); 2D d d ta c:
( ) ( ) ( ) ( ) ( )
( )
3;13. 0 1 3 3 1 0
1 1; 1
DdDA DE d d d d
d D
== + =
=
.
VDvFnm khc pha so viAEnn ta c ( ) ( ) ( )1; 1 5; 1 ; 1;5D C B .
Vy ( ) ( ) ( )1;5 ; 5; 1 ; 1; 1B C D .
Cu 5. Gii hphng trnh( ) ( )
2
5 3 2 2 3 4
2 2
x y x y x y x y x
y x
+ + = + + +
+ =
Li gii.
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Tham gia Luyn thi mn TONti MOON.VNt im scao nht trong k thi THPT Quc gia 2016
iu kin20;2 0
3 0;3 4
y x
x y x
+
Phng trnh thnht ca htng ng vi
( ) ( )
( ) ( )
( )
2 5 2 3 4 3 0
2 12 2 1 0
3 4 31
2 1 2 03 4 3
x y x y x y x x y
x yx y x y
x x y
x y x yx x y
+ + + + + =
+ + =
+ +
+ + = + +
V1
2 0 2 13 4 3
x y y xx x y
+ + > = + +
. Phng trnh thhai ca htrthnh
Khi phng trnh thhai ca htrthnh ( )22 1 2 2 1x x + = .
p dng bt ng thc lin htrung bnh cng trung bnh nhn ta c
( ) ( )
( )
2 22 2
222
1 2 1 1 2 3 22 1 2 1 2 1 1 2
2 2 24 13 2 42 2 1 2 2
2 2 2
x x x xx x x x
xx xx x
+ + + + = + + =
+ = = +
Do phng trnh (1) c nghim khi cc du ng thc xy ra, tc l2
2 1 11
2 1
xx
x
= =
=.
i chiu iu kin, kt lun hv nghim.
Cu 6. Trong mt phng vi htoOxycho tam gicABCvung cn tiAvi Ml trung im caAB,ng thng quaAvung gc viMCctBCtiH, bit phng trnh ng thng : 1 0AB x y + =
v trung im caHBl 145; 3K . Tm tocc nh ca tam gic
ABC bitBc honh ln hn 4.
Li gii:
Ta c:2
2 4.5
CI AC CI CM AC
CM CM
= = =
.
DthyMKl ng trung bnh ca tam gicABHkhi ta
c / /MK IHdo vy4
5
CH
CK= v HK BK= .
Do vy 5KC BK = . Ta c:
( )0 4
; sin 45 3d K AB KB KB= =
Gi ( ); 1B t t+ ta c: ( )
( )
( )
222
5 5;611 16
5 11 11 143 9 ;
3 3 3
t B
KB t t t B loai
=
= + = =
Khi : ( )
5 6.0
6 5; 2146 6. 6
3
C
C
x
BC BK Cy
=
= =
Phng trnh ng thng qua Cv vung gc viABl 3 0x y+ =
Do vy ( )1;2A AB AC A= .Kt lun: ( ) ( ) ( )1;2 ; 5;6 ; 5; 2A B C l cc im cn tm.
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Cu 7: Gii hphng trnh
2 2 2
33
2 4 2 3 4 2
3 2 12 1 3 2 2 1
2
x y x y y
x yx y x y
+ + + + = + + + +
+ = +
Li gii.
iu kin 31 2
;
2 3
x y . Phng trnh thnht ca htng ng vi
( )( )
( ) ( )
2 2 2
2 2 2
2 2 2
2 2 2 4 3 4 0
2 2 2 4 3 4
10
1 1 1 2 1
x y x y y
x y x yx y
x y x y y
x yx y x y
x y x y y
+ + + + + + =
+ +
+ + + + + + +
+ + = =
+ + + + + + +
Phng trnh thhai ca htrthnh3
3 3 2 12 1 3 2 12
x xx x x
+ + = + .
p dng bt ng thc Cauchy ta c3 3 3
3 2 1 1 3 2 1 3 2 1 3 2 12 1 3 2 12 2 2 2
x x x x x xx x x
+ + + + + + = + .
Do phng trnh nxc nghim khi cc du ng thc xy ra, tc l 32 1 3 2 1 1x x x = = = .Cu 8. Trong mt phng vi h ta cho tam gic ABC vung ti B c phn gic trong AD vi
15 1;
2 2D
thuc BC .Gi E, F l 2 im ln lt thuc cc cnh AB v AC sao cho AE AF= . ng
thng EF ct BC ti K. Bit im11 3
;2 2
F
, E c tung dng v phng trnh ng thng
: 2 1 0AK x y + = . Tm tocc nh ca tam gicABC.Li gii:
GiI l giao im caAD vEF . Do tam gicAEF cn tiA cphn gicAI nn:AI l phn gic ng thi l ng cao v trungtuyn.
Ta c:KE AD
DE AKAB KD
. Do ng thngDE qua
15 1;
2 2D
v vung gc viAK. Khi ta c phng trnh
31: 2 02DE x y+ = .VE thucDE nn ta gi 31; 22E t t
Dthy ( )2
21515 2 5
2DE DF t t
= + =
( )
( )
( )
2
17 17 3;
2 2 22 15 4
13 13 5; 6;2 : 8 0
2 2 2
t E loai
t
t E I AD x y
=
=
= + =
Khi ( )5;3 :3 18 0; : 3 14 0; : 3 22 0A AD AK A AC x y AB x y BC x y= + = + = =
Do vy ( ) ( )20
5;3 ; 8;2 ; ; 23
A B C
l tocc im cn tm.
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Kha hc Chinh phc Hnh phng Oxy v Kthut gii Hphng trnh Thy ng Vit Hng Moon.vn
Tham gia Luyn thi mn TONti MOON.VNt im scao nht trong k thi THPT Quc gia 2016
Cu 9: Gii hphng trnh
( ) ( )
2 2 2 2
2 2
2 33
5 6 5 6 8 6
2 4
x x y
x xy y x xy y
x y x x y
++ = + +
+ + =
Li gii.
iu kin
0
0 22
0
x
x y xy x
y
Nhn xt( ) ( ) ( )
( ) ( ) ( )
2 2 2 2 22 2
2 2 22 2 2 2
5 6 55 6 5 4
6 8 6 5 6 8 6
x xy y x y x yx xy y x y x y x y
x xy y x y x y x y x xy y x y x y
+ + = + + = + + +
+ = + + + + + = +
Dn n( )
2 2 2 2
32 3 2 33
5 6 5 6 8 6
x yx x y x x y
x y x y x yx xy y x xy y
++ ++ + = =
+ + + + +
.
ng thc xy ra khi v chkhi x y= . Phng trnh thhai trthnh
( ) ( ) ( )2 1 2 024 1 1
0 0
x xx xx x x x
x x
+ =+ = + = = =
.
Kt lun hphng trnh c nghim duy nht 1x y= = .
Thy ng Vit Hng