3. CaĆ¢u ĆaĆ¹p aĆ¹n ĆieĆ„m
7
(1,0Ʊ)
GoĆÆi E vaĆø F laĆ n lƶƓĆÆt laĆø giao ƱieĆ„m cuĆ»a HM vaĆø HG
vĆ“Ć¹i BC. Suy ra
āāā
HM =
āāā
ME vaĆø
āāā
HG = 2
āāā
GF,
Do ƱoĆ¹ E(ā6; 1) vaĆø F(2; 5).
0,25
A
B C
!
D
H
#
M $
I
%
G
E '
F
ĆƶƓĆøng thaĆŗng BC Ʊi qua E vaĆø nhaƤn
āāā
EF laĆøm vectĆ“
chƦ phƶƓng, neĆ¢n BC : x ā 2y + 8 = 0. ĆƶƓĆøng thaĆŗng
BH Ʊi qua H vaĆø nhaƤn
āāā
EF laĆøm vectĆ“ phaĆ¹p tuyeĆ”n, neĆ¢n
BH: 2x + y + 1 = 0. ToĆÆa ƱoƤ ƱieĆ„m B thoĆ»a maƵn heƤ
phƶƓng trƬnh
x ā 2y + 8 = 0
2x + y + 1 = 0.
Suy ra B(ā2; 3).
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Do M laĆø trung ƱieĆ„m cuĆ»a AB neĆ¢n A(ā4; ā3).
GoĆÆi I laĆø giao ƱieĆ„m cuĆ»a AC vaĆø BD, suy ra
āā
GA = 4
āā
GI. Do ƱoĆ¹ I 0;
3
2
.
0,25
Do I laĆø trung ƱieĆ„m cuĆ»a ƱoaĆÆn BD, neĆ¢n D(2; 0). 0,25
8
(1,0Ʊ)
(1 ā y)
ā
x ā y + x = 2 + (x ā y ā 1)
ā
y (1)
2y2
ā 3x + 6y + 1 = 2
ā
x ā 2y ā
ā
4x ā 5y ā 3 (2).
ĆieĆ u kieƤn:
ļ£±
ļ£²
ļ£³
y ā„ 0
x ā„ 2y
4x ā„ 5y + 3
(ā).
Ta coĆ¹ (1) ā (1 ā y)(
ā
x ā y ā 1) + (x ā y ā 1)(1 ā
ā
y) = 0
ā (1 ā y)(x ā y ā 1)
1
ā
x ā y + 1
+
1
1 +
ā
y
= 0 (3).
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Do
1
ā
x ā y + 1
+
1
1 +
ā
y
0 neĆ¢n (3) ā
y = 1
y = x ā 1.
ā¢ VĆ“Ć¹i y = 1, phƶƓng trƬnh (2) trĆ“Ć» thaĆønh 9 ā 3x = 0 ā x = 3.
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ā¢ VĆ“Ć¹i y = x ā 1, ƱieĆ u kieƤn (ā) trĆ“Ć» thaĆønh 1 ā¤ x ā¤ 2. PhƶƓng trƬnh (2) trĆ“Ć» thaĆønh
2x2
ā x ā 3 =
ā
2 ā x ā 2(x2
ā x ā 1) + (x ā 1 ā
ā
2 ā x) = 0
ā (x2
ā x ā 1) 2 +
1
x ā 1 +
ā
2 ā x
= 0
0,25
ā x2
āx ā1 = 0 ā x =
1 Ā±
ā
5
2
. ĆoĆ”i chieĆ”u ƱieĆ u kieƤn (ā) vaĆø keĆ”t hĆ“ĆÆp trƶƓĆøng hĆ“ĆÆp treĆ¢n, ta ƱƶƓĆÆc
nghieƤm (x; y) cuĆ»a heƤ ƱaƵ cho laĆø (3; 1) vaĆø
1 +
ā
5
2
;
ā1 +
ā
5
2
.
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9
(1,0Ʊ)
Ta coĆ¹ a + b + c ā„ 2 a(b + c). Suy ra
a
b + c
ā„
2a
a + b + c
. 0,25
TƶƓng tƶĆÆ,
b
a + c
ā„
2b
a + b + c
.
Do ƱoĆ¹ P ā„
2(a + b)
a + b + c
+
c
2(a + b)
=
2(a + b)
a + b + c
+
a + b + c
2(a + b)
ā
1
2
0,25
ā„ 2 ā
1
2
=
3
2
. 0,25
Khi a = 0, b = c, b 0 thƬ P =
3
2
. Do ƱoĆ¹ giaĆ¹ trĆ² nhoĆ» nhaĆ”t cuĆ»a P laĆø
3
2
. 0,25
āāāāāāHeĆ”tāāāāāā
3