Phương pháp giải toán hình học theo chuyên đề part 3

Phucmg
phdp gidi Todii
Hinh
hoc theo chuyen de - Nguyen Phu
Khdnh,
Nguyen Tat Thu
1
xTir;
2)
Taco
A'N
=
-^A'B'=^N
B'Q
=
-B'Br:>Q
4
4
4
^a\f3 a
3aV2
,
AT
=
-A'C'=>P
3
2
2 4
va M
0;a;
0;^;aV^^

Suy ra
AN
=
f
aVs
AQ =
4
4
aS
a
3aV2'
;^;a^/^
, AM
=
0;a; —
,AP
=
2 2 4
AP,AQ
AP,AQ
0;
.AN
=
-^a3;
24
AP,AQ
AM
=
-
Dodo

VA.MI,Q=7[AP,AQ'
.AN
AI^AQ
AM
=
aV6
IF
w
w w w 13a^N/6
VAMPNQ
-
VA.MI'Q
+
VA.MPQ
- -•—— •
£Q BAI
TAP
Bdi
1.1.3,
Trong khong gian Oxyz cho cac vec to:
a
=
(2;3;l),
b
=
(-3;-2;0), c
=
(x;2;-3)
1)
Tim

X
de a
A
b vuong goc
voi
c
2)
Tim
X
de goc giua hai vec to a
A
b va c
bang
120".
Jiuang
dan
gidi
Ta
CO
aAb =
(2;-3;5)
1)
aAblco(aAb).c
=
0c>2x-3.2
+
5.(-3)
=
0«x
=

y
(aAb).c
- 1
^
'
•
= cosl20°=
2)
Yeu cau bai toan o
a AB
2x-21
1
=
—
<=>
Vx^+13.V38
2
21
^21
x<
—
2
2(2x-21)2
=19(x^+13)
x<
2 o x =
llx2+168x-635
=
0
-84

±
V14041
11
Cty
TNHHMTV nVVII Khang Vict
P^i 2.1 •3-
Trong khong gian vai he tpa dp Oxyz cho diem
A
(3;-2;
4)
f
im
tpa dp cac hinh chieu ciia A len cac true tpa dp va cac mat
phSng
toa dp.
2j Tir"
^ ^ ^ *a"^ 8'^*^
AMN
vuong can tai A
3^
Tim
tpa dp diem E thupc mat
phang
(Oyz) sao cho tam
giac
AEB can tai E
CO
dien
tich
bang

3^y29 voi
B(-1;4;-4)
. ,^
Jiucang
dan
gidi
\
Gpi
Ai,A2,A3
Ian lupt la hinh chieu cua A len cac true Oa, Oy, Oz.
^'"^
chieu cua A len cac mat
phang
tpa dp (Oxy),(Oyz),(Ozx).
Taco: Aj
(3;0;0),A2(0;-2;0),A3(0;0;4)
1; "
va
Bi(3;-2;0),B2(3;0;4),B3(0;-2;4)
.y.
2)
Do
M€Ox=>M(m;0;0),
N€Oy=i>N(0;n;0) F' '
Suyra
AM =
(m-3;2;-4),
AN =
(-3;n
+

2;-4)
Tam
giac
AMN
vuong can tai A nen ta c6 •
-3(m-3)
+
2(n
+ 2) + 16 =
0
2(n
+
2) +
16
AM.AN
=
0
AM^
= AN^
(m
-
3)^ + 2^ +
(-4)2
=
(-3)2
+
(n
+
2)^ +
(-4)^

m-3
=
2(n + 2) +
16
3
n2
(1)
=
(n
+ 2)2+5
(2)
Ta
c6: (2)
o
4(n +
if
+
64(n
+
2) + 256 = 9(n +
if
+
45
o5(n +
2)2
-64(n
+
2)-211
=
0

n
+ 2
=
n
+ 2
=
32
+
3V23T
5
32-37231
<=>
n
=
•
n
=
•
22 +
3N/23T
5
22-3^/23T
m
=
-
m
=
189 +
67231
5

189 -
67231
'
l%y
CO
hai bp thoa yeu cau bai toan:
loSc
M,
M^
'189 +
67231
15
f 189 -
67231
15
;0;0
;0;0
^^^22
+ 37231^^1
0;
22-37231
;0
Phuortig
phap giai Todn
Hinh
hgc theo chuyen de- Nguyen Phu
Khdnh,
Nguyen Tat Thu
3)
Vi Ee(Oyz) nen E(0;x;y)

Suyra
AE =
(-3;y+ 2;z-4),
BE =
(l;y-4;z
+
4)
=
(8y
+
6z-8;4z
+
8;10-4y)
AE,BE
Nen
tir gia thiet bai toan ta c6:
AE^ = BE^
12
AE,BE
=
3sl29
<=>
AE^ = BE^
AE,BE
AE^ = BE^
o9
+
(y
+
if

+
(z-4)^
=
1
+
(y-4)^
+{z
+
if oy
=
=
1044
4z
+
l
AE,BE
'50z-16
=
1044 « (8y
+
6z -
8)^ + (4z + 8)^ + (lO
- 4yf
=
W44
2
•(4z
+
8)'
^26-16zf_,044

=
0«z.2,z ^
25
•
z
=
2::^y
= 3
nen E(0;3;2)
34 37 . ^
•
z
=
=>y
=
nen E
25 ^ 25 V 25 25;
Bdi
3.1.3.
Trong khong gian Oxyz cho bon diem
A(l;
2; 1), B(2; -1; -3), C(-2;
0;
3), D(0;
3; 4)
1)
Chung
mlnh
rkg
A,

B,
C,
D khong dong phang.
Tinh
the
tich
ciia
tu
dif
n
ABCD
2)
Tinh
chieu cao ve tu
B
ciia
tam giac
BCD
va chieu cao
ciia
tii
dien
ABCD
ve
tu
A.
3) G(?i M,
N Ian lugt la trung diem
ciia
AB

va
CD.
Tinh
c6 sin
ciia
goc
giiia
hai
duong thang
CM
va
BN.
4) Tim E
tren duong thSng
AB
sao cho tam giac
ECD
c6 di^n
tich
nho nhat.
Jiuang
dan
gidi
l)Tac6
BC =
(-4;1;6),
BD =
(-2;4;7),
BA =
(-1;3;4)

Suyra
BCABD
=
(-17;16;-14),
(BCABD)BA
= 99^0
Vay
A,B,C,D
khong d6ng phSng va the
tich
khoi
tu di?n
ABCD
la:
6
(BCABD)BA
=1 (dvtt).
2)
Taco:
CD = (2;3;l):r>CD =
7l4,S^BCD
=:
Suy ra duong cao
BB'
=
=
^.
BCABD
V74T
•

214
Cty
TNHH
MTV DWHKhangVift
3V
Chieu
cao
ciia
hinh chop ve tir A
la:
h =
^ABCD
3)
Ta CO
M
^•1-1
2'2'
3 7]
~''2'2j
-
-
- ^7 1
Suy ra CM= -;-;-4
V
2
2
,BN:
Nen
cos(CM,BN)-
CM.BN

•
5 13^
^'2'
2
141
•
CM.BN
=
V24T'
141
4)
Ta
CO
phuong
trinh
AB:
CM.BN
276555
x =
l
+
t
y =
2-3t, suy ra
E(l +
t;2-3t;l-4t)
z
=
l-4t
CE

=
(t
+ 3;
-3t
+ 2;
-4t -
2),
CE
A
CD =
(9t +
8;
-9t
- 7;
9t +
5)
Suy ra
S^CDE^^
CEACD
=
i\/243?
+
360t
+
138
2
1
2^
243
t +

20f
14 ^ >/42
+
— >•
3 6
Vay
S^( Q£ nho nhat
khi
va chi khi E
27,
f
7 ,38,107'
27'
9 ' 27 ,
Bai
4.1.3. Trong khong gian
Oxyz,
cho ba diem
A(l;l;l),
B(5;l;-2),
C(7;9;l).
1)
Chung
minh
rang cac diem
A, B,
C khong thang hang.
2)
Tim
toa do diem D sao cho

ABCD
la hinh binh hanh.
3)
Tinh
cos
A,
sin
B,
tan C
ciia
tam giac
ABC.
4)
Tinh
do dai duong cao tu dinh
A,
ban
kinh
duong tron ngoai tiep, ban
kinh
duong tron noi tiep
ciia
tam giac
ABC.
5) Tim
toa dg giao diem
ciia
phan giac trong, phan giac ngoai goc A
v6i
duong

thSngBC.
Jiuang
dan
giai
1)
Ta
CO
AB(4;0;-3),AC(6;8;0)
nen cac diem
A, B,
C thang hang khi va chi khi
[4 =
6k
ton
tai so thyc k sao cho
AB = kAC,
tiic
la
V%y
A,
B,
C khong thang hang.
0
=
8k (v6
li).
-3
= O.k
215
Phumigphdpgidi

Todit llhih hoc llico clim/eit ile- Ngni/On Phil Khi'iiih, Njjuylit Tat Thu
2) Vi A, B, C khong thing hang nen ABCD la hinh binh hanh khi va chi khi
DC = AB, hay
9-yD=0 «
1-ZD=-3
XD=3
yD=9=>D(3;9;4).
ZD=4
r
•
.'1.
^
3)Tac6 AB = +
0^
+ (-3)2 =5, AC =10, BC
=777.
Vay
cosA
=
cos(AB,AC)
=
AB.AC
.24_12
•
50 ~ 25•
AB
AC
Tuongtu
cosB
=

^,cosC
= ^^^. Vi
B,C€(0;7r)
nen
385 385
+)
sinB>0=>sinB
2^ 2 481
=
Vl-cos
B =-,
5
V
77
+)
tanC,cosC
cung dau tanC = .|—I 1 =
kos^C
38
4) Dien tich tam
giac
ABC la S
=-BA.BC.sinB
=-5.N/77 j—=
VisT.
2 2 5
V
77
Do do
.) h, =

2S^2S
a BC V 77
+) R =
= 2
AC
'481
2sinB 2sinB
= 25.
77
r
= ^ =
2S
481
2V48T
p
AB + BC + CA 15 +
V77*
Chii
y: Co the
tinh
dipn
tich bang cong thuc S =
AB,AC
5) Goi E, F Ian lugt la
giao
diem ciia phan
giac
trong, phan
giac
ngoai goc A voi

duong
thSng
BC.
Theo
h'nh
cha't
phan
giac,
ta c6 — = — =
^ ^ EC FC AC
+) Vi E nam trong doan BC nen EC = -2EB
Giasu
E{x^;y^;z^) thi EC(7 -
XE;9
-
yj:;l
-
),EB(5-
XE;1
- y^;-2-z^)
Nen EC = 2EB tuong duong voi
01
<;
Cty
TNIIH
MTV DVVll Khang Vi?t
7,XE
=-2(5-XE)
9_y£=-2(l-yE) «
1 ZE=-2(-2-ZE)

17
Xc
=•
11 ^fl7 11 ,
ZE=-1
Tuong tu, do F nam ngoai doan BC nen FC = 2FB, tu do ta tim
dugc
tga
^^diem
F la
F(3;-7;3).
gai 5'
1
Trong khong gian Oxyz, cho tam
giac
ABC c6
A(2;3;l),B(-l;2;0),C(l;l;-2).
1) Tim toa do
chan
duong vuong goc ke txx A xuong BC
2) Tim tga do H la true tam cua tam
giac
ABC.
3) Tim tga do I la tam duong tron ngoai tie'p cua tam
giac
ABC.
4) Gpi G la trgng tam cua tam
giac
ABC. Chung minh rang cac diem G, H, I
nam tren mot duong thing.

Jiumigdangiai
'.r j; •
1) Ggi K la
chan
duong vuong goc ke tu A xuong BC.
x =
-l
+ 2t
Ta
CO
BC = (2;
-1;
-2), phuong
trinh
BC:
<
y = 2-t
z = -2t
Suy ra K(2t -1; 2
-1;
- 2t), AK(2t - 3; -1
-1;
-1 - 2t)
Vi
AK1 BC AK.BC = 0 o (2t - 3).2 + (-1 - t).(-l) + (-1 -
2t).(-2)
= 0
« t = i. Tga do diem K can tim la K
M 5 2^
3 3 3)

2) Ggi H(x;y;z) la tryc tam tam
giac
ABC. Ta c6
AH(x-2;y-3;z-l),BH(x + l;y-2;z)
AB(-3;
-1;
-1), AC(-1; - 2; - 3), BC(2;
-1;
- 2)
Tich
CO
huong cua hai vec to AB, AC la
r
n
-1
-1
-1
-3
-3 -1
\
AB,AC
-1
-1
•
-1
-2
AB,AC
\
-2 -3 -3 -1
-1

-2
/
=
(l;-8;5).
Vi
H la true tam tam
giac
ABC nen
'"it' VSi
217
Phucntgphapgiai Todn
Hinli
iiQC
theo
chuyen de N^iiijcti I'hu
Khanh,
Nguyen Tat
Thu
AH±BC
BHICA
o
H
6
(ABC)
AH.BC
= 0
BH.CA
= 0
AB,AC
.AH

= 0
2x-y-2z = -l
x + 2y + 3z = 3
x-8y + 5z = -17
Giai
h§ ta dugc H
^
2 29
1
^
{15
15 3)
3)
Goi
I(x;y;z)
la tam duong tron ngoai tiep tam giac
ABC.
Ta c6
AI(x-2;y-3;z-l),BI(x
+ l;y-2;z),a(x-l;y-l;z + 2).
Vi
I la tam duong tron ngoai tiep tam giac ABC nen
AI^
=
BI^
AI^
=
CI^
o
•

AB,AC1.AI
= 0
AI
=
BI
AI
=
CI
IG
(ABC)
6x + 2y + 2z = 9
x + 2y + 3z = 4
x-8y + 5z = -17
Giai
ta dup-c I
61 1^
^15 30 3,
4)
Trpng
tam G cua tam giac
ABC
c6 toa dp thoa man
2-1 +
1
3 + 2 +
1 1
+ 0-2^
2
•2-1
,3'''

3)
Do do HG
115 15
,GI
^;
-1;0
15 30
nen HG =
2GI,
tiic la ba diem G,
H,
I nam tren mot duong thSng.
Bai
6.1.3.
Cho tam giac deu ABC c6 A(5;3;-l),B(2;3;-4) va diem C nam
trong m^t phang (Oxy) c6 tung do nho hon 3.
1)
Tim tpa dp diem D biet
ABCD
la tu
di?n
deu.
2)
Tim tpa dp diem S biet
SA,SB,SC
doi mpt vuong goc.
Jiuong
dan
giai
Vi

Ce(Oxy)
nen
C(x;y;0).
Ta
CO
AB(-3;
0; -
3),
AC(x
- 5;y -
3;
1),BC(x
- 2;y - 3;4)
Tam
giac ABC la tam giac deu nen AB =
AC
=
BC,
do do
(x-5)2+(y-3)2+1^=18
(x-5)2
+(y-3)2 +1^ =(x-2)2 +(y-3)2 +4^
X =
1;
y = 4
x = l |x = l;y=2'
AC
=
AB
|AC=BC'

|(y-3)2=1
218
Cty
TNHl!
MTV nVVH Kluuix Viet
Vi
C
CO
tung dp nho hon 3 nen
C(l;2;0).
l)Gpi
D('<;y;z)- ' '
Khi
do AD(x - 5;y - 3;z +
l),BD(x
-2;y -3;z +
4),CD(x
-1;y - 2;z)
Tam
giac ABC la tam giac deu nen
ABCD
la tu difn deu khi va chi khi
=
BD = CD = AB =
3N/2
.
Ta c6 h§ phuong
trinh
[(X
- 5f + (y - 3)2 + (z +1)2 = (X -

2)2
+ (y -
3)2
+ (z + 4)2 .^J'
(x -
5)2
+ (y -
3)2
+ (z +1)2 = (X -1)2 + (y -
2)2
+ z2
•
1
:ir:).'f:>
ivu
'ij
I ' ,f .(/.
(x-5)2+(y-3)2+(z + l)2=18
z
= l-x
y = 16-5x
(x-5)2+(13-5x)2+(2-x)2=18
z
= l-x
y-16-5x
3x2-16x +20 = 0 V' ' •
x = 2
x = —•
10 _2 _7
3'

3' 3
Giai
phuong
trinh
3x2 - 16x + 20 = 0
^j^^^^
Vay
tpa dp cac diem D la D(2; 6; -1) hoac D
2)Gpi
S(x;y;z).
Taco:
AS(x-5;y-3;z + l),BS(x-2;y-3;z +
4),CS(x-l;y-2;z)
SA,
SB,
SC doi mpt vuong goc khi va chi khi
.•;'A
.fc' )
it
AS.BS
= 0
(X
•
BS.CS
= 0
<=>
J
(X
CS.AS
- 0

(X
^x2+y2+z2-
7x
o
•
x2+y2+z2-
3x
x2+y2+z2-
6x
y = 5z + ll
o
<
x = l-z
3z2+10z + 8
=
=
0
x + y-4z = 12
-3x - 3z = -3
x2
+ y2 + z2 - 6x - 5y + z = -11
Giai
phuong
trinh
3z2 +
lOz
+ 8 = 0 ta dupe z = -2;z = ' ''^^'
219
Phucmgphapgidi
Todn limit UQC

theo
chtiySn de-
Nguyen
Phi'i Khanh,
Nguyen
Tat Thu
^7 13 4~
Suy ra hai diem S thoa man la
S(3;1;-2),S
3' 3
Bdi
7.1.3.
Trong khong gian Oxyz, cho hinh hpp chu nhat ABCD.A'B'C'D'
CO
A = 0,B€0x,D€0y,A'G0z va AB =
1,
AD = 2, AA' = 3.
1) Tim toa dp cac
dinh
ciia hinh hop.
2) Tim diem E tren duong thang DD' sao cho B'E 1 A'C
3) Tim diem M thuoc A'C, N thupc BD sao cho MN 1BD,MN 1 A'C. Tu
do
tinh
khoang
each
giua hai duong thang
cheo
nhau A'C va BD .
Jiucang

ddn
gidi
1)
Taco
A(0;0;0),B(1;0;0),
D(0;2;0),
A'(0;0;3).
Hinh
chieu ciia C len (Oxy) la C, hinh chieu cua C len Oz la A nen
C(l;2;0).
Hinh
chieu cua B',C',D' len mp(Oxy)va true Oz Ian
lupt
la cac diem
B,C,D va A' nen
B'(l;0;3),
C'(l;2;3),
D'(0;2;3).
2) Vi E thupc duong thang DD' nen E(0;2;Z) , suy ra B'E =
(-l;2;z-3)
Ma A^ =
(l;2;-3)
nen
B'ElA'C«B^.A^-0
»-l +
4-3(z-3)
= 0oz = 4. Vay
E(0;2;4).
3) Dat A/M = x.A^; BN = y.BD
Ta

CO
AM =
AA^
+ A?M =
AA;
+
X.A^
=
(x;2x;3
-3x),
suy ra M(X;2X;3-3X)
AN
= AB + BN = AB + y.BD = (l - y;2y;0) => N(1 - y;2y;0)
Theo
gia thiet cua de bai, ta c6:
H.
MN.A'C
= 0
MN.BD
= 0
Ma MN = (l-x-y;2y-2x;3x-3),A'C = (l;2;-3), BD =
(-1;2;0)
Nen
(*) <r>
fl
- x - y + 4y - 4x - 9x + 9 = 0 [-14x + 3y = -10
-l
+ x + y +
4y-4x
= 0

-3x + 5y = l
X =
y =
53
61
44
61
Do do M
f53.106.24^
61'
61 '61,
Ml 17 88 ^
Vi
MN la duong vuong goc chung ciia hai duong thang A'C,BD nen
220
Cty TNHH MTV DWH Khang Viet
d(A'C,
BD) = MN = ^{1-x-yf + (2y-2xf + {3x - sf =
6761
61
gai
8.1.3.
Trong khong gian v6i he true toa dp Oxyz cho hinh chop
S.ABCD
CO
day ABCD la hinh thang vuong tai A, B voi AB = BC = a; AD = 2a ;
A
= 0,B thupc tia Ox, D thupc tia Oy va S thupc tia Oz. Duong thang SC va
BD tao vai nhau mot goc a thoa
cosa

=
-
^
So'
V
1) Xacdjnhtpa dp cac
dinh
cua hinh chop
(;••;(•
2) Chung minh rang ASCD vuong,
tinh
dien tich tam
giac
SCD va
tinh
c6 sin
cua goc hpp bai hai mat phang (SAB) va
(SCD).
3) Gpi E la trung diem canh AD. Tim tpa dp tam va
tinh
ban
kinh
mat cau
ngoai tiep hinh chop
S.BCE
.
4) Tren cac canh SA, SB, BC, CD Ian
lupt
lay cac diem M, N, P, Q thoa SM =
MA,

SN = 2NB, BP =
3PC,CQ-4QD
. Chung minh rang M, N, P, Q khong
dong
phang va
tinh
the tich kho'i chop MNPQ. ' ' ' '
Jiuong
ddn
gidi
1)
Taco
A(0;0;0),B(a;0;0),D(0;2a;0),C(a;a;0).Dat
SA = x
=>S(0;0;x)
^ BD = (-a; 2a; 0), SC - (a; a; -x)
:r>
DB = a V5, SC =
Vx^
+ 2a2;
BD.SC
- a^
a 1
Nen
cosa
=
COS(SC,BD)
SC.BD
SCBD
J5(x2+2a2)

N/30
o x^ + 2a^ = 6a^ o
X
- 2a
S(0;0;2a).
2) Ta
CO
CS =
(-a;-a;2a),CD
=
(-a;a;0)
CS.CD
= 0 =>
ASCD
vuong tai C.
Do do:
S.crn
=
^CS.CD
- i.aV6.aV2 = a^v/s
'ASCD
Gpi
p =
((SCD),(SAB)).
Ta
CO
hinh chieu cua tam
giac
SCD len mat phSng (SAB) la tam
giac

SAB nen ta suy ra
1
S ^^-23 1
cosP
=
|^=2^=
1 . N
^ASCD
a yl3 V3
3)
Taco
E(0;a;0).
Gpi
l(x;y;z) la tam mat cau
^ ngo^i tiep hinh chop
SBCE
' x
77i
Phuang phapgiai Toa,i Hinh
hoc
theo chuyen
dj-
NguySn Phii Khdnh,
Nguyen
Tat Thu
=
-
a)2 + y2
+ ^
^2 ^ y2

^ _
2a)2
Khi
do J
IC2
= «
(X
-
a)2 + (y
-
a)2 + z2 =
x2
+ y2
4-
(2
- laf
. , ,
I
lE^
=IS'
-2x
+ 4z =
3a
-x-y + 2z =
a<=>
-2y
+
4z
=
3a

x^+(y-a)2+z2=x2+y2+(z_2a)2
a
X = —
2
z=:a
^a
a ^
Ban kinh
R =
IE
=,
4)Tac6
M(0;0;a)
Do
SN = -SB=>N
3
v2y
a
a
—
V
2y
+
a
2
a
iV6
CQ
= JCD=^Q
r2a

I
3 '°' 3
5'
5'"
1
3 —
,
BP = -BC=>P
a;
) 4
V
4
J
Suy
ra
MN
=
3
••"•'-i
\
MP
=
( 3a ^
a;—-;-a
4
,MQ
=
a
9a
a^

a^^
MN
A
MP
=
•
4 ' 3 ' 2
nen M, N,
P, Q
khong dong phang"!
(MNAMP).MQ
=
—.^0
VMNPQ
=^|(MNAMP).MQ
40
Bdi
9.1.3.
Trong khong gian
v6i h^ tpa dp
Oxyz
cho
hinh
hpp chu
nhat
ABCD.A'B'C'D'
c6 A
triing
voi goc tpa dp,
B(a;0;0),D(0;a;0)

,A'(0;0;b)
voi
(a
>
0,b >
O). Gpi
M la
trung diem cua CC.
1)
Tinh
the
tich cua khoi
tu
dien
BDA'
M
.
2)
Cho
a
+ b =
4.
Tim max
V^.gPi^
.
Jiu&ng
ddn
gidi
1)
Taco: C(a;a;0),

B'(a;0;b),
C'(a;a;b),
D'(0;a;b)
z:>M(a;a;|)
Suyra ]VB =
(a;0;-b),
]VD =
(0;a;-b),
A'M=
a;a;
Cty
TNHH
MTV
DVVII
Khang
Vigt
pgn
A'B,A'D
=(ab;ab;a2)
=>A'M.rA'B,A'D
Vay
V^'MBD
= —•
2) Do a,b
>
0
nen
ap
dung
BDT

Co si
ta c6:
4
=
a
+ b
=
ia
+
ia
+
b>33/iX:^a2b<~
2
2 V4 27
Sa^b
DSng
thiic
xay ra <=>
^
=
b
2
<=>
a +
b
=
4
8
a-
—

3
64
4
•
Vay
maxV^.BDM
= 27
^~3
Bai
10.1.3.
Trong khong gian Oxyz
cho tu
di^n
deu
ABCD
c6
A(3;-l;2)
va
7'
G(l;l;l)
la
trpng tarn
tam
giac
ABC.
Duong thang BC
di qua M
djnh
tpa dp
cac

dinh con lai
va
tinh
the
tich khoi
tu
dien
do.
Jiuang
ddn
gidi
4;4;-
Taco:
AG
=
(-2;2;-l),
AM = (l;5;y)
Suy ra
nABC
= AG
A
2AM = (-12; -24; -24)
Phuong trinh (ABC)
:x
+
2y
+
2z-5
=
0.

Ug(; = AG A
nABC
= (6; 3; -6)
Phuong trinh
BC:
x
= 4 +
2t
y = 4 +
t
z =
-^-2t
2
Gpi
a
la canh ciia
tu
di?n
ABCD,
ta c6
AG =
—
=
3
=>
a = 3N/3
2)
Xac
4
+

2t;4
+
t; 2t
2
Ta
c6: B
Nen
ta
dupe phuong trinh:
AB
=
2t
+
l;t
+
5;-2t- —
2
(2t
+ l)2+(t + 5)2
+ 2t +
Buy
ra B
2)
4-S
I
+
2S]
-27
ot2+4t
+

—=
0<»t
= -2±
,
C
7i;
4 +
>/3
Phuoiigphdp
gidi
Todn Hinh
hoc
theo chuyen di-Nguyen Phii
Khdnh,
Nguyen
Tat
Thu
X
= l
+
t
Do DG
1
(ABC)
nen phuong trinh GD
J
y
=
1
+

2t
j
z =
l
+
2t
Suy
ra
D(l
+
t;l
+ 2t;l + 2t)
' A'
Ma
GD =
VDA2-AG2
=
—=
372=^3
t
=3V2=>t
=
±^y2
3
Dodo
S(l
+ V2;l +
2V2;l
+ 272) hoac
S(1-^;1-2V2;1-2V2).

Thetichcua tudien:
V
^lDGS.r,r
=-—.^-^
=
(<lvtt).
§ 2. LAP
PHl/ONG
TRlNH
MAT
PHANG
iDe lap phuong trinh mat phang (a),
ta
c6 cac each sau:
Cdch l:Tim
mot
diem
M(X(,;yg;zo)
ma mat
phang
(a) di qua va mpt
VTPT
n
=
(a;b;c). Khi
do
phuong trinh ciia (a) c6 dang:
a(x-Xo)
+
b(y-yo)

+
c(z-Zo)
= 0.
Mot
so'
luu
y
khi tim VTPT cua mat phang (a):
•
Neu hai
vec
to a,b
khong ciing phuong
va
c6
gia
song song hoac nam tren
(a)
thi
a
A
b
= n
la
VTPT cua (a).
•
Neu mat
phang
(a) di
qua

ba
diem phan biet khong thang hang
A, B,
C thi
AB
A
AC
= n
la
VTPT cua (a).
•
Neu
(a)//(P)
thi
1^
=
1^.
•
Neu
A1
(a)
thi n,^ = u^ .
•
Neu
(a)l(P)
thi
n,^//(a).
•
Neu
A(a; 0;

0), B(0; b; 0), C(0;
0; c)
voi
abc
*
0
thi phuong trinh (ABC):
abc
Cdch
2: Gia su
phuong trinh
(a) c6
dang: ax
+
by
+
cz
+
d
=
0
Dxfa vao
gia
thiet
cua de
bai
ta
tim
dugc
ba

trong bo'n
an
a,
h,
c,
d
theo
3'^
con
lai.
Chang
han a = mb, c = nb, d = pb. Khi do
phuong trinh
(a)
mx
+ y +
nz
+ p
—
0.
Cty
TNHH MTV DWII Khang Viet
Chii
I/:
Neu
mat
phang
(a) di qua
M(xo;yo;Z(j)
thi

phuong trinh
cua (a) c6
aang:
a(x-Xo)
+
b(y-yo)
+
c(z-Zo)
=
0.
i
fidu 1.2.3. Lap phuong trinh mat phang (a), biet:
1)
(a)
di
qua ba diem A(1;1;1),B(2;-1;3),C(-1;2;-1),
2) (a)
di
qua hai diem A,
B
va song song voi OC
3)
(a) di qua
M(l;l;l), vuong
goc
voi
((3):
2x
-
y

+
z
-1 =
0
va
song song
vaiA—-j-^,
4) (a) vuong goc voi hai mat phang (P):x+y + z-l = 0, (Q):2x—y+3z-4 = 0
va khoang each tir O den (a) bang
^26 .
JCffigidi.
1) Taco AB
=
(l;-2;2), AC
=
(-2;l;-2), suy
ra
ABA AC
=
(2;-2;-3)
Phuong trinh (a): 2x
-
2y
-
3z
+
3 =
0 .
2) Taco
OC-(-l;2;-l),suyra

ABAOC-(-2;-1;0)
Vi (a)
di
qua
A, B
va
song song voi
CD
nen (a) nhan
n =
-
AB
A
OC
=
(2;1;0) lam VTPT.
Suy
ra
phuong trinh
(a):
2x
+
y
—
3 = 0 .
3) Taco: i^-(2;-l;l),
S^
=
(2;l;-3)
Do

(a)l(P)
.(a)//A
=>-a=n,An,=(2;8A).
Phuong trinh (a):
x +
4y
+
2z
-
7
=
0.
4) Taco
iv,'
=
(l;l;l), n^
=
(2;-l;3) Ian lugt la VTPT cua
(P) va (Q).
Vi (a) vuong goc voi hai mat phang
(P) va (Q)
nen (a) nhan vec
to
n
=
A
n^
= (4;
-1;
-3) lam VTPT.

Suy
ra
phuong trinh (a) c6 dang :4x-y-3z +
d=:0.
^Mat
khac:
d(0,(a))
=
V26 nen
ta
c6:
i
= N/26
^
d
=
±26.
726
I
Vay phuong trinh (a):
4x
-
y
-
3z
± 26 =
0.
^^idu
2.2.3.
Lap phuong trinh mat phang

(P),
biet:
1) (P)
di
qua
giao tuyen ciia hai mat phiing
(a):
x-3z-2=0; ([3):y-2z+l=0_
225
Phuang
phdpgiai ToAn
Hinh
hoc
theo
chuyen
de -
Nguyen
Phu
Khdnh,
Nguyen Ta't
Thu
va khoang each tir
M
den
(P)
bang
6^3
'
2)
(P) di

qua hai diem A(];2;1),B(-2;1;3)
sao cho
khoang each tu C(2;-l;i)
den
(P)
bang khoang each tu
D
(O; 3; 1)
den (P).
Lai
giai.
1) Gia sir
(P):
ax
+ by + cz + d =
0.
:h
Taco
A(2;-1;0),B(5;1;1)
la
diem chung eua
(a) va (3)
Vi
(P) di
qua
giao
tuyen ciia hai mat phlng
(a) va (|3) nen A,B e (P)
Suy
ra

2a-b
+ d = 0
5a
+ b +
c
+ d = 0
b
=
2a
+ d
c 7a-2d'
1
.
—
e + d
2
c
+
2d|
=
Va^
+
b^
+
e^
«
27(e
+
2d)^
=

49{a^
+ +c^)
3V3
27.493^
= 49
a2+(2a
+ d)2+(7a +
2d)2
a
= -d
o27a2+32ad
+ 5d2 =0o
a
=
-—d
27
•
d =
-a=>b
= a;c =
-5a.
Suy ra
phuang
trinh
(P) la:
ax
+ ay -
5az
- a = 0ox +
y-5z-l

= 0.
27
17 36
•
d = a => b = a;c = a. Suy ra
phuang
trinh
(P) la:
5
5 5
5x-17y-36z-27-0.
2) Gia sir
(P):
ax
+ by + cz + d 0
Vi A,Be{P)^-
a+2b+c+d=0
-2a
+ b +
3c
+ d = 0
<=>s
a
=
•
d
= -
-b
+
2c

3
5b
+
5c
Mat khac:
d
(C,
(P)) = d
(D,
(P))«|2a
- b + c +
d|
=
|3b
+ c + d
<»|5b-c|
=
|2b-c|oc
=
|b,c
= 0
•
Voi
c-|b=>a
=
2b;d^-yb^(P):4x
+ 2y +
7z-15
= 0
226

Cty
TNHH
MTV DWH
Khang Viet
IVai
b =
0=>a
=
-c;d
=
c=>(P):2x
+ 3z-5 = 0.
3
3
fidu
2.2.3. Trong khong gian
Oxyz
cho diem
A(l;2;3)
va
hai duong
,
x-1 y + 1 z + 2 , x-2 y + 2 z-1
L"!"'-]
d
1 : —
/
cl 9
: = , .
'2

1 -1 1 2-4 'I
rU-n.^
u-:-i
.••
j) Viet phuong
trinh
mat phclng
(P)
di qua
A va dj. I
2) Chung
minh
rang
di va di cat
nhau.
Viet phuong
trinh
mat
phang
(Q)
chua
di va
d2.
JCffigidi-
Taco:
Duong th^ng
di di
qua M(l;-l;-2),
VTCP
u^

=
(2;1;-1)
Du6ngthangd2diqua
N(2;-2;l),
VTCP
u^
=
(l;2;-4)
1)
Taco:
AM =
(0;-3;-5)
Do
(P) di
qua
A va dj nen
n^
= AM A
u^
=
(-8;10;-6)
Suy
ra
phuong
trinh
(P):
4x
- 5y + 3z -
3
= 0.

2)
Xet he
phuong
trinh
l
+
2t
=
2
+ t'
-l
+ t =
-2
+
2t'o
-2-t
=
l-4t'
Suy
ra di va d2
cat nhau tai E{3;0;-3).
2t-t'
= l
t-2t'
=
-l <=>t-t'
= l
-t
+
4t'

= 3
Ta CO
ng = Uj A
U2
=
(-2;
7;
3)
Phuong
trinh
(Q): 2x - 7y - 3z
-15
= 0 .
ViduL 4.2.3. Trong khong gian
Oxyz
cho
ba
duong thang
,
x-1 y+1 z-1 . x+1 y-1 z , ,
dj
: =
^
= , dj : = - = — va
d^
:
1
2 -1^2 3-1 ^
x
=

-2t
y
=
-l-4t.^,,,
.
z
=
-l
+
2t
"
1) Viet phuong
trinh
mat phSng
(a) di qua d2 va cat
di,
ds
Ian lugt tai
A, B
sao cho
AB =
N/IS
.
2)
Goi (P) la mat
phang chua
di va di. Lap
phuang
trinh
mat

phang
(Q)
chua
d2 va
tao
voi
mat ph^ng
(P)
mot
goc
(p thoa cosip
=
^) Ta
CO
A 6 dj =^ A(l +
a; -1
+
2a;
1
-
a), B
e
da
=>
B(-2b; -1
-
4b; -1
+ 2b)
Suy
ra

AB
=
(-a-2b-l;-2(a
+
2b);a
+
2b-2),
dat x = a + 2b
STU AB
=
>/13=>(X
+
1)2+4X^
+(X-2)^=13OX
=
-1,X
= -
Phuong phapgiai Todn Hinh
hqc
theo chuyen
de-
Nguyen
Phu
Khdnh, Nguyen
Tat
Thu
•
Voi x =
-l=>
AB =

(0;2;-3),
ta CO
u =
(2;
3;
-1) la
VTCP
cua d2
va
A(-l;
1;
0)
e
=>
A e
(a)
Suy ra
n =
[AB, il]
=
(7; -6; -4) la
VTPT
cua
(a).
Phuong
trinh
(a): 7x
-
6y
-

4z +13
=
0.
4
f 7 8 2^1
• Voix
=
—=>AB=
—;—;— .
3
V 3 3 3J
Suy ra
n =
[-3AB,u]
=
(-14;11;5)
la
VTPT
cua
(a).
Phuong
trinh
(a): 14x -1 ly
-
5z
-
25
=
0.
2)Du6ngthang

dj
di
qua M(l;-l;l) c6
VTCP
u^
=
(l;2;-l)
Duong thang
dj
di
qua E(-1;1;0), c6
VTCP
u^
=
(2;3;-l)
Duong
thing
dj
di
qua
N(0;-1;-1)
vacoVTCP
u^
=
(-2;-4;2)
Taco
u^
=
-2u7=>di//d3.
Dodo

II7
=
U^AMN
=
(-4;3;2)
Vi
mat phang
(Q)
di qua
6.2
nen phuong
trinh
(Q)
c6 dang:
ax
+
by
+
cz
+
a
-
b
=
0
(1)
voi a^ + b^
+
c^
>

0
va
2a
+
3b
-
c
=
0 c
=
2a + 3b.
|4a-3b-2c|
9 b
Mat khac cos
9 =
Nen ta c6:
np.ng
—•—.
np
HQ
9b
^2%a^
+ +c^)
729(5a2+12ab
+
10b2)
6
V29(5a2+12ab
+
10b2)

^
o
3^/5
|b|
= 2V5a2+12ab + 10b2
o
IQa^ + 48ab
-
5b^
= 0
<=>
a = -^b
10
a
= -Sb
2
•
a
=
—b
ta chpn
b =
10 =>
a
=
l,c
=
32
.
Phuong

trinh
(Q)
la:
x
+
lOy
+
32z
-
9
=
0.
•
a =
b
tachon
b =
-2=>a
=
5,c=:4.
2
Phuong
trinh
(Q)
la: 5x
-
2y
+
4z
+

7
=
0
.
Vi
du
5.2.3. Trong khong gian
Oxyz
cho
duong thang
A c6
phuong
trinh
x-l_y+l
z+3
-2
va diem M(l;2;0).
Cty
TNHH
MTV DWH
Khang Vi?t
Viet phuong
trinh
mat phang
(a)
di
qua M, song song voi
A
va
(a)

tao voi
ba tia Ox, Oy,
Oz
mot tu dien c6 the
tich
bang
8.
2) Viet phuong
trinh
mat phang
(P)
di qua M, vuong
goc
voi
(P):x+y+z-3=0
[31
va t^o voi
A
mot
goc
(p thoa cos
(p
= /— . ,,,1
V 34
Xffi
giai.
I) Gia su
(a)
cat ba true Ox, Oy,
Oz

Ian lugt tai
A
voi
a,b,c>0
Khi
do, phuong
trinh
(a): ax
+
by
+
cz
=
1
Vi
(a)
di
qua
M
va
song song voi
A
nen ta c6:
fa
=
-2b +
l
-;0;0
f
1 •

,B
0;-;0
V
b
,C
0;0;-
c
fa + 2b
=
1
i
<=>
<,
-2a + 2b + 3c =
0
c
=
-2b +
-
3
Mat khac
VQABC
=
8
<=>
-OA.OB.OC
=
8
<=> abc
= —

o b(2b
-
l)(2b
- -) =
J-
«
192b3
-
160b^
+
32b -1
= 0
3'
48
,1
11
b
=
-=>a
= -,c
=
-
4
2 6
1
1
1'
I
,j
o (4b

-
l)(48b2
-
28b +1)
= 0
^ 7
+
V37 5-V37 1-N/37
b
=
=> a
=
,c
=
24 12
12
^ 7-V37 5
+
V37
1
+
V37
b
=
=>a
=
,c
=
24
12 12

Vay phuong
trinh
mat can lap la: (aj): 6x
+
3y
+
2z -12
=
0
;
(a2): 2(5
-
v/37)x
+
(7
+ V37)y +
2(1
- V37)z -
24
= 0
Va (a3):2(5+>/37)x
+ (7-V37)y
+
2(l
+
V37)z-24
=
0.
'
2)

Vi
mat phang
(3)
di
qua
M
nen phuong
trinh
ciia
(3)
c6
dang:
/ '
*
a(x-l)
+
b(y-2)
+
cz
=
0^ax
+
by
+
cz-a-2b
=
0
(*) . (^)'.; ' ^
voi a^
+

b^+c^ >0.
•
I Mat khac,
(P)
1 (P)
nen
rfp.rip
=
0oa
+ b +
c
=
0c=>c
=
-a-b.
Taco:
sin(p= sin(u^,np)
u^.np
|5a + b|
. n,
|-2a
+
2b +
3c|
Vr7.Va2+b2+c2
" 734(77^b2)
Phuontg
phap
giai
Toan Hinh hoc

theo
chuyen
de-
Nguyen
Phu Khdnh,
Nguyen
Tat Thu
. f: — [3" - . '
\5a
+ b [3"
Ma
sin
(D = vl - cos cp =
J—
nen ta co: , =, —
, V34
V34(a2+ab
+ b2) V34
(5a + hf = 3{a^ + ab + ) « 22a^
+
7ab -
Zb^
= 0 »
• a = —b,
tachon
b^ll=>a
=2,c = -13
nen
phuong
trinh

cua (p) la: 2x + lly - 13z- 24 = 0.
• a = -^b,
tachon
b =
-2=>a
=],c = l
nen
phuang
trinh
cua (3) la: x - 2y + z + 3 = 0 = 0.
11
a = -lb
2
Vi
du
6.2.3.
Trong khong gian Oxyz cho hai mat phang
(P):x+2y+2z-3=0,
x-4 _y _ z
mat phang (Q): 2x - y + 2z - 9 = 0 va duong thang ^' ^ ^ 2
1) Gpi (a) la mat phang phan
giac
cua goc hop boi hai mat phang (P) va (Q).
Tim
giao diem cua duong thang
A
va mat phSng (a).
2) Viet phuong
trinh
mat phSng (3) di qua giao tuyen ciia (P) va (Q), dong

thoicach
E(8;-2;-9)
mot khoang Ion nhat.
1) Goi M la giao diem cua mat phang (a) va duong thSng A.
Taco
MGA
nen M(4 + t;t;-2t)
Vi
M e (a) nen ta c6: d(M,(P)) = d(M,(Q))
|4
+ t +
2t-4t-3|
|2(4 + t)-t-4t-9| , „ , „
—^ = ^—
^o|t-l|
= |3t +
l|<::>t
= 0,t = -l
o o
Tu
do ta
CO
duoc hai diem M la: Mi(4;0;0) va M2(3;-l;2).
2) Ta
CO
A(3;-l;l) va
B(-4;l;9)
la hai diem thuoc giao cua (P) va (Q).
Do do (3) di qua giao tuyen cua hai mat phang (P) va (Q) khi va chi khi
A,B€(3).

fx
= 3-7t
Taco
AB =
(-7;2;8)
nen phuong
trinh
AB:
Gpi
K la
hinh
chieu cua E len AB, suy ra
K(3 - 7t;
-1
+ 2t;
1
+ 8t)
=>
EK = (-5 - 7t;
1
+
2t; 10 + St)
y
=
-l
+ 2t, t€]
z = l + 8t
Cty
TNHH MTV DWH Khang Viet
Vi

EK 1 AB => EK.AB = 0 -7(-5 - 7t) + 2(1 + 2t) + 8(10 + 8t) = 0 » t = -1
Suyra
K(10;-3;-7),
EK =
(2;-1;2).
Goi
H la
hinh
chieu cua E len mat phang (3), khi do:
d(E,(P))
= EH < EK
Suy ra
d(E,(3))
Ion nhat khi va chi khi H = K hay (3) la mat phang di qua
va vuong goc voi EK , ; ,
Phuong
trinh
(p): 2x - y + 2z - 9 = 0 . (ta thay (P) = (Q)).
Vi
du
7.2.3.
Trong he toa do Oxyz, cho hai duong thang t, ,
x-1 y-2 z-4 X y-3 z-2 , , ^ , nx
di
•.
— = ^^ = —^,d2 :- = ^-j- = — vadiem A(-2;l;0) .
Chung
minh
A,di,d2
cung nam trong mot mat phang. Tim toa do cac

dinh
B,C cua tam
giac
ABC biet duong cao tir B nam tren dj va duong phan
giac
trong goc C nam tren 62 .
JCgigidi.
Duong
thang di di qua M(l;2;4) vac c6 VTCP
u,"
=
(1;1;1)
Duong
thang d2 di qua
N(0;3;2)
vac c6 VTCP u^ = (l;-l;2) * '
Goi
I la giao diem ciia
di,d2
=> I(l;2;4) " ' '
Mat
phang (a)
chua
di, d2 c6 n^ =
Ui,U2
=(3;-l;-2)
va di qua I nen
jphuongtrinh
:3x-y-2z
+ 7 = 0.

Ta thay A e (a). Vay
A,di,d2
ciing thuoc mp (a)
Xac djnh diem C:
Goi
(3) la mp di qua A va vuong goc voi dj => (p): x + y + z +1 = 0
Co C = (P) n
(d2)
nen tpa do diem C la nghiem cua h^ phuong
trinh
:
x+y+z+l=0 • ;
'
x_y-3_z-2^C(-3;6;-4)
1
-1 2
Goi
A' la diem do'i xung cua A qua d2 .
Ke
AHi.d2
=>H(t;3-t;2 + 2t) => AH = (t +
2;2-t;2
+ 2t)
AH.U2
-0o(t +
2)-(2-t)
+ 2(2 + 2t) =
0r:>t
•H
2.n.2

'3' 3 '3j
A'
2.19.4
3' 3 '3)
Co duong thang BC la duong thang BA' di qua C va c6 VTCP
= CA' = i(ll;l;16) chon u' = (ll;l;16)
3
Phuangphdpgiai
Todn
Hhih
hoc
theo
chuyen
de-
Nguyen
Phu
Khdnh,
Nguyen
Tat Thu
c
u .^'uo/-x + 3 y-6 z + 4
Suv ra phuang
tnnh
BC: = = .
^ ^ ^ 11 1 16
Tpa dg diem B la nghiem cua he phuang
trinh:
'x + 3_y-6_z + 4
^29.34.44^
I

5 ' 5 ' 5 ,
11
1 16
x-1
_ y-2 _ z-4
B
. Vay B
29 34 44
5'5'5
;C(-3;6;-4)
1 1 1
du
Trong khong gian
Oxyz
cho duong thang
x-4m
+ 3 y-2m-3 z-8m-7
dm :
—I ::— = :— = —•_ 1— voi m i
-1-1-1
' 4'2
2m-l
m +
1
4m + 3
Chung
minh
rang khi m thay doi thi duang thang
dm
luon nam trong

mot
mat
phang co'djnh. Viet phuong
trinh
mat phang do.
Vm
XgigidL
Duong
thing
dm
di qua A(4m - 3;2m +
3;
8m + 7) va c6
VTCP
u
= (2m -1;m + l;4m + 3)
Gia su dm c (a): ax + by + cz + d =
0
voi moi m, khi do ta c6:
f
a(2m -1) + b(m +1) + c(4m + 3) = 0
|a(4m-3)
+ b(2m + 3) + c(8m + 7) + d-0
•(2a + b + 4c)m -
a
+ b + 3c = 0
o { Vm
[(4a + 2b + 8c)m - 3a + 3b + 7c + d = 0
•2a
+ b + 4c = 0

-a + b + 3c = 0
2a + b + 4c = 0
-3a + 3b + 7c
+
d = 0
Ta chpn a =
1
=> b = 10,c - -3,d = -6
V|y dm
c(a):x
+ 10y-3z-6-0.
Vidu
8.2.3.
Trong khong gian
Oxyz
cho bon diem A(l;l;l),B(-l;0;-2),
C(2;-l;0),D(-2;2;3).
1) Chung
minh
rang A, B, C, D la bon
dinh
cua mot tu
difn
va
tinh
the
tich
tu
di^n ABCD.
2) Lap phuang

trinh
mat phang (a) song song voi AB, CD va cat hai duang
b
= 10a
c = -3a
d
= -6a
thang
AC, BD Ian lugt tai hai diem M, N thoa
BN
AMj
= AM^
-1.
3)
GQI
G la trong tam cua tu dien
ABCD,
(P) la mat phgng di qua G cat ca£.
Cty
TNHH
MTV DWH
Khang
Viet
AB,
AC, AD Ian lugt tai
B',C',D'.
Viet phuang
trinh
mat phang (?)
jji^'t

tu dien
AB'C'D'
c6 the
tich
Ian nhat.
Xgigidi.
1)
Ta c6: AB = (-2;
-1;
-3), AC =
(1; -2;
-1), AD = (-3;
1;
2)
Suy ra AB A AC = (-5; -5; 5) (AB A
AC).AD
= 20 ^ 0
Nen A, B, C, D la bo'n
dinh
cua tu dien.
The
tich
tu
dien ABCD la: V,XBCD = - (AB A
AC).AD
6
10
'Sr.''
2) Ta CO
BD

=
(-1;2;5)
nen
BD
=
N/30,
AC = 4^
CD = (-4;
3;
3) => AB A CD =
(6; 18;
-10)
Vi
(a) song song vai
AB,
CD nen n„ =(3;9;-5). '
Vi
MN,
AB,
CD Ian lugt n5m tren ba mat phang song song gom: Mat phang
di
qua AB va song song vai
CD,
mat phang di qua CD va song song voi AB va
mat
phang (a) nen thco
dinh
li Talet dao trong khong gian ta suy ra:
BN AM BN AM /p , „.,2 r 2
= o-= =<=> BN =

N/5AM
<:> BN = 5AM
BD
AC N/6
Mat khac:
BN
^
VAM
Ta CO phuong
trinh
AC :
= AM' -1 <^ AM"* - AM^ = BN^ = SAM^ ^ AM^ = 6
x
= l + t
y
= l-2t=^M(l +
t;l-2t;l-t)
z = l-t
2
Suy ra AM = (t;-2t;-t) => AM^ = 6ot = ±l.
t
= 1, ta CO M(2;
-1;
0) = C nen ta loai truong hgp nay.
t
=
-1,
ta CO M(0;
3;
2) nen phuang

trinh
(a): 3x + 9y - 5z -17 = 0.
3) Ta CO G
2 2
Ggi Ai la trong tam cua tam
giac
ABC, suy ra Aj
f_i.i.r
. 3'3'3.
Ian.;-,"
\
SuyraAG=
-1;
—; , AAj =
V 2 2)
' AG = - AAV = 1(AB + AC + AD) = 1
4 4^ ^4
4 _2 _2
•3'
3' 3)
. Tu do ta c6:
AC AD'
^AB'
Vi
B',C',D',G dong phSng nen ta c6:
4
1
AB AC_ AD
AB'^ AC'^ AD';
, AB AC AD .

=
1
=> + + = 4.
AB' AC AD'
2331
Phuang
phdpgiai
Todn
Hinh
hoc
tlteo
chuycn
de - Nguyen Phu
Khdnh,
Nguyen
TatThu
Ap
dung bat
dling
thuc Co si ta c6:
4 =
^
+
^
+
^>33/AB7AC
AD
^
AB;
AC

AD'^
27
AB'
AC AD'
"VAB''AC'AD'
^
AB
' AC '
AD
^ 64
,u'
VAB'CD'
AB'.AC.AD'^
27 ^27^, 45
L
.u' • lu— ui^u- AB AC AD 3
^ang thuc xay ra
khi
va
chi khi
= =
=
—
^
^ AB' AC AD' 4
Do do: V^BCD'
"ho nhat
<=>
(a) song song
voi

mat phang
(BCD).
Ta c6: BC = (3;
-1;
2), BD = (-1; 2; 5) => n;;^ = BC A BD = (-9; -17; 5)
Vay
phuong
trinh
mat phang (a) la: 9x +
17y
- 5z -
6
= 0 .
m
BAI TAP
Bdi
1.2,3.
Viet
phuong
trinh
mat phSng
(a),
biet:
x
= 2t
1)
(a) di qua
diem
A(2;3;-l)
va duong thang dj : y =

1 -1,
[z
=
2
+ t
X
+
1
y
—
2
z
2)
(a) chua hai duong thang
di
va dj
:
——
=
^—^—
= —,
3)
(a) chua
di
va song song
voi
Oy.
Jiicang
dan gidi
1)

Duong thSng
di
di qua
B(0;l;2),
VTCP
L^ =
(2;-1;1)
Suy ra
AB =
(-2;-2;
3),
n^
= AB
A
u^ = (1;
8;
6)
Phuong
trinh
(a): x +
8y
+
6z -
20 =
0.
2)
Duong thang
d2
di qua
C(-l;2;0),

VTCP
u^
=
(-2;l;-l)
Suy ra
BC = (-1;
1;-2)
r:.
n^
= u7 A BC = (1;
3;
1)
Phuong
trinh
(a):
x +
3y
+
z -
5 =
0.
3)
Oy
CO
VTCP
k
=
(0;1;0).
Suy ra
ri^ =

uj
A
k
=
(-1;0;2)
Phuong
trinh
(a):
x
- 2z
=
0.
Bdi
2.2.3.
Trong
khong gian
Oxyz
cho 4
diem
A(l;l;l),
B(-l;2;0),
C(-2;0;-l),D(3;-l;2).
1)
Viet
phuong
trinh
mat
phing
(ABC).
Tim

tpa dp
tri^c
tam tam giac
ABC.
2)
Viet
phuang
trinh
mat phang (a)
di
qua
AB
va song song
voi
CD
3)
Viet
phuang
trinh
mat phang (p) di qua
AB
va
each
deu C, D
4)
Viet
phuang
trinh
mat
phling

(P) di qua
BC
va
each
A mpt khoang Ian nhat.
Cty
TNHH
MTV DWH
Khang
Vift
Jiucarig
dan
gidi
^)
Taco
AB =
(-2;l;-l),A(:
=
(-3;l;2)^nABC=(3;7;l)
phuongtrinh
(ABC):
3x +
7y
+
z-11
=
0
Goi
H(a;b;c) la
true

tam tam giac
ABC,
ta c6:
\
H
e
(ABC)
CH.AB
=
00
BH.AC
=
0
3a
+
7b
+ c-ll = 0
2a-b
+ c + 5 = 0
<=>•
3a-b-2c
+ 5 =
0
a =

b =
c
=
59
127

59
-24
59
.Vay
H
72 127 24
"59'
59 ' 59
2)
Taco
CD =
(5;-l;3):::>n„
=ABACD =
(2;1;-3)
Phuong
trinh
(a):
2x
+
y - 3z
= 0
.
3)
Phuong
trinh
(P)
c6 dang: *
a(x-l)
+
b(y-l)

+
c(z-l)
= 0
voi a^ +b^ +c^
5^0
te^
Do B €
(a) nen suy ra -2a
+
b-
c =
0=>c
=
-2a
+
b
H
Nen ta
viet
lai
phuang
trinh
(P)
nhu sau: ax
+
by
+
(-2a
+
b)z

+ a
-
2b =
0
Vi
(a)
each
deu
C,
D nen
d(C,(a))
=
d(D,(a))
V
o |a -
3b| = |b|
o a
=
4b,a
=
2b
B a
=
4b,tachon b
= 1 =>
a
= 4 .
Phuong
trinh
(p):4x

+
y-7z
+ 2 =
0
B a
=
2b,tachon b
=
l=>a
=
2.Phuangtrinh
(P):2x
+
y-3z
=
0.
•
[x=::-l
+
t
y = 2 + 2t
_ „
4)
Ta
CO
phuang
trinh
BC:
z
=

t
Goi K
la
hinh
chieu cua A len
BC,
suy ra
K(-l +1;2 + 2t; t)
AK =
(t-2;2t
+
l;t-l).
Do
AK.BC
=
0=>t
=
-=i>AK
=
6
11
4
"6'3'
5^
6,
Gpi
H la
hinh
chieu
eiia

A len mat phSng
(P),
ta c6
AH =
d(A,(P))
Va AH < AK
nen
d(A,(P))
Ian nhat
khi
va
chi khi
H
= K
Hay
(P) la mat phSng di qua
K
va
vuong
goc
voi AK
Phuongtrinh
(P):llx-4y
+
5z
+ 19 =
0. J,
231
Phucmg phdp giai
Todn Ilinh

hQC
theo chuyen de - Nguyen Phu
Khdnh,
Nguyen Tat Thu
Bai 3.2.3. Trong khong gian Oxyz cho ba duong thang
d
.
X
_ y _ z-l ^ _x-l_y + l_z + 2 ^ x + l_y_z + l
1) Viet phuang trinh mat phang (a) di qua A(1;2;3),B(-1;0;2) va cat d^d^
Ian lugt tai C, D sao cho CD
=
VsS .
2) Viet phuong trinh mat phang (p) song song va each deu hai duong thang
di va ds.
3) Viet phuang trinh mat phang (P) di qua O va cat di, ds Ian lugt tai hai diern
M, N sao cho MN
=
\/l4 dong thai MN song song vai mat phang
(Q):2x + y + z-l = 0 va <-^.
Jiuang d&n giai
1) Ta
CO
duong thSng di c6 VTCP
u^
=
(2;-l;-2),
duong thing
d2
c6 VTCP

U2
=
(-2;!; 2) va duong thang ds c6 VTCP
=
(1;1;1).
C e dj =^
C(2c;
-c;
1
-
2c),
D € dj D(l - 2d; -1 + d; -2 + 2d)
CD
=
(-2x
+
l;x-l;2x-3),
x =
d
+
c
Suyra CD
=
VsS (2x -1)^ + (x -1)^
+
(2x -
3)^
- 38
^9x2-18x-27
= 0<:>x = -l,x = 3

• X = -1
=>
CD =
(3;
-2;
-5), AB = (-2;
-2;
-1)
Suy ra n„ =
AB
A
CD
=
(8;-13;
10)
Phuang trinh (a): 8x - 13y
+
lOz -12
=
0
•
x-3=>CD-(-5;2;3).Suy
ra n„ =
ABACD
= (-4;11;-14)
Phuang trinh (a): 4x -1 ly
+
14z - 24
=
0.

2) Vi (p) song song voi hai duong thing di,
da
nen ta c6
Hp
=Ui
AU3
=(l;-4;3)
Phuong trinh (P) c6 dang: x-4y
+
3z
+
m
=
0
Lay E(0;0;l)6di,
F(-l;0;-l)6d3
Vi (P) each deu hai duong thang d^,d^ nen ta c6:
d(E,(P)) = d(F,(P))<=> m + 3 = m-4
1
o m = -
2
V^y phuang trinh
(P):
x
- 4y
+
3z + ^
=
0.
Cty

TNHH MTV DWH Khang Vift
3) Ta CO M(l - 2m; -1 + m; -2
+
2m), N(-l +
n; n;
-1 + n)
Suy ra MN
=
(n
+
2m-2;n-m
+
l;n-2m
+
l)
Do MN//(Q) nen MN.nQ = 0 -:>r
c>
2(n
+
2m - 2)
+
(n - m +1)
+
(n - 2m +1)
=
0 « m - 2 - 4n • -
Matkhac: MN = Vl4 o(n
+
2m-2)^+(n-m+
1)^+(n-2m+

1)^
=14
92
^ 155n2 _ 92n
=
0
<^ n
=
0,n
=
— . ,^ ^ ,
po
XN<-^
nentaco n
=
0
^ m
= 2
M(-3;1;2),N(-1;0;-1)
Suy ra n7 = OMAON =
(-l;-5;l)
' .^p;;
Phuang trinh (P):
X
+
5y -
z =
0
.
Bdi

4,2.3.
Lap phuang trinh mat phing (a) biet '^^
1/.
1) (a) qua hai diem A(l; 2; -1), B(0; -3; 2) va vuong goc voi mat phang
(P):2x-y-z
+
l
=
0.
2) (a) each deu hai mat phang
(p):
x
+
2y-2z
+
2 =
0,
(y):2x + 2y
+
z
+ 3 = 0. '
3) (a) qua hai diem C(-l;0;2),D(l;-2;3) va khoang each tu goc tga dg tai mat
phang (a) la 2. / ;
4) (a) song song voi mat phSng (Q):
x
- 2y - 2z -
3
=
0
va khoang each giiJa

hai mat phang la 3.
5)
(a) di qua
E(0;
1;
1)
va d(A,(a)) = 2;d(B,(a)) = y, trong do A(l;
2;
-1),
B(0;
-3;
2).
Jivccmg dan gidi
l)Tae6
AB(-l;-5;3),ii(p)(2;-l;-l)
nen
|_AB,
n(p)
J
=
(8;5;ll).
Mat phang (a) qua A,B va vuong goc vai mat phang (P) nen
'AB,ii(p)]
= (8;5;ll).
fi(a) ^ "(a) ^
"(P)
^
"(a)
=
Phuong trinh mat phang (a) can tim

8(x-l) + 5(y-2) + ll(z + l)-0o8x
+
5y + llz-7
=
0.
2) Goi M(x;y;z) la mot diem bat ky thugc mat phing (a).
x
+ 2y-2z
+
2|
_|2x
+
2y
+
z
+
3|
r
Taeo
d(M,(P))
=
d(M,(y))«>
237
Phumig
phdp
gidi
Todn
Hinh
hgc
theo

chuyen
de-
Nguyen
Phil Khdnh,
Nguyen
Tat Tliu
x + 3z + l=0
3x + 4y-z + 5 = 0
» x +
2y-2z
+ 2 = 2x + 2y + z + 3
"x + 2y - 2z + 2 = 2x + 2y + z + 3
X + 2y - 2z + 2 = -2x - 2y - z - 3
Vay
CO
hai mat phang (a) can tim la
(a):
X
+ 3z +1 = 0
hoac
(a): 3x + 4y - z + 5 = 0.
3) Mat phang (a) di qua diem
C(-l;0;2)
nen c6 phuong
trinh
dang
A(x
+1) + By + C(z - 2) = 0, + B^ + > 0.
Vi(a)qua D(l;-2;3) nen 2A - 2B + C = 0 =^ C = 2B - 2A (1).
A-2C

Ta
CO
d(0,(a)) = 2 nen = 2 (2).
The (1) vao (2) roi
binh
phuong,
riit
gon ta thu du^c
'A
= 2B
5A^
-8AB-4B^
=0c=>
2 . Do A^ + B^ + > 0 nen
A
= B
5
• Neu A = 2B thi chon B =
1
=> A = 2,C = -2, do do phuong
trinh
mat phang
(a) la 2x + y-2z + 6 = 0.
2
• Neu A = — B thi chon
13
=-5 => A = 2,C =-14, do do phuong
trinh
mat
phang (a) la

2x-5y-14z
+ 30 = 0.
Vay
CO
hai mat phang thoa man 2x + y-2z + 6 = 0,
2x-5y-14z
+ 30 = 0.
4) (a) song song voi mat ph3ng (Q):
x-2y-2z-3
= 0 nen c6 phuong
trinh
(a):
X
^ 2y - 2z + D = 0. Lay diem
N(3;0;0)
e (Q).
3 + D
Ta
CO
d((a),(Q)) = d(N,(a))« = 3
Vl^+(-2)2+(-2)2
D
+ 3
=:9oD
= 6;D = -12.
o
Vay
CO
hai mat phang can tim x - 2y - 2z + 6 = 0, x - 2y - 2z -12 = 0.
5) Mat phSng (a) qua E(0; 1; 1) c6 phuang

trinh
dang
Ax
i
B(y-1)
+
C(z-1)
= 0, A^ + B^+C^ >0.
Theobaira
d{A,(a))-2;d(B,(a))-y
voi A(l;2;-l),B(0;-3;2) nen
CtyTNHH MIV D VVHKhang Vi?t
A
+ B-2C
VA^
+ B^+C^
-4B + C
= 2
11
A
+ B -
2C|
= 2VA^ +B^ (1)
11
A + B-2C
=14-4B
+ C (2)
VA^
+ B^+C^ 7
11(A

+ B-2C) =
14(-4B
+ C)
Tir
(2) ta c6
11(A + B-2C) =
14(4B-C)
A
=
A
=
-67B + 36C
11
45B + 8C
11
A -67B + 36C ,
• Vol A = — , thay vao (1) ta co phuang
trmh:
11
r-56B
+
14C
11
= 4
-67B + 36C
11
\
+
B2+C^
<=>

3826B2
-
4432BC
+
1368C^
= 0 (3) '
•
Phuang
trinh
(3) chi c6 nghiem B = C = 0, khi do A
:=
0 (khong thoa man
dieukifn
A^ + B^+C^ >0)
• Voi A =
-^55^1^^
tiiay
vao (1) ta c6 phuong
trinh
56B-14C
11
11
\2
= 4
45B + 8C
11
+ B^ +
o
13626^
+

1112BC
+ 136C^ = 0 <^ B
= C,B
=-—C.
3 227
2
+) Vol B = C thichpn C =
-3=>B
= 2,A = 6 phuang
trinh
mat phang (a) la
3 I
6x +
2y-3z
+ l = 0.
+) Voi B = -—C thichQn C = 227 =>B
=-34,A
= 26 phuong
trinh
(a) la
26x - 34y +
227z
-193 = 0.
Vay
CO
hai mat phSng can tim la: i' i j '
6x +
2y-3z
+ l = 0,
26x-34y

+
227z-193
= 0.
Bai
5.2.3.
Trong khong gian Oxyz cho mat cau ^ •
(S): x^ + y^ + z^ - 4x + 6y - 2z - 28 = 0 va hai duang thang
. x +
5_y-1_z
+ 13 , x + 7 _ y + 1_ z-8
• 2 -3 2 ^ 3 -2 1 '
239
Plunrii^
phdp ^iai loan Iliiili hoc
Uieo
chuycn dc -
N^itiicn
I'hii KIti'uih,
Nguyen
Tat Thu
1) Viet phuang
trinh
mat phSng (P) tiep xuc voi mat cau (S) va song song v6,
hai
duong thang di; d2.
2) Viet phuang
trinh
mat phang (a) di qua di va tao voi d2 mot goc cp thoa
V42
COS(p

= .
7
3) Viet phuang
trinh
mat phang (P) chua di va cat (S) theo mot ducmg
tron
c6
ban kinh
bang ^^^^.
Jiuang
dan
gidi
Mat cau (S) c6 tam 1(2; -3; 1), ban
kinh
R = 742
Duong thSng dj di qua Mi(-5;1;-13),
VTCP
Uj"
=
(2;-3;2)
Duong thSng 62 di qua
M2(-7;-l;8),
VTCP
u^ =
(3;-2;l)
1) Vi mat phling (P) song song voi hai duong thang dpdj nen ta c6:
np
=ui AU2
=(1;4;5)
Suy ra phuang

trinh
(P) c6 dang: x + 4y + 5z + m = 0
Mat khac, (P) tiep xuc voi (S) nen
d(I,(P))
= R
m
-5
742
= 742 <=> m-5 = 42om = 47,m = -37
Vay phuang
trinh
(P) la: x + 4y + 5z - 37 = 0
hoac
x + 4y + 5z + 47 = 0 .
2) Vi mat phang (a) chiia dj nen phuong
trinh
(a) c6 dang
a(x + 5) + b(y -1) + c(z +13) = 0 <=> ax + by + cz + 5a - b + 13c = 0
Voi
a^+b^+c^
^0 va 2a-3b + 2c = 0«b = ^^^^
3
Taco:
sin(p =
7l-cos(p
=-j=, suy ra cos|n„,U2
77
"a "2
1
3a - 2b + c

- =
——<z>
77
, 2a + 2c
3a-2. + c
14.Va^+b^+c^
77
n
a2 +
r2a + 2c^
+ c
o(5a-c)2
=2(13a2+8ac
+
13c2)<=>a2+26ab
+ 25c2 =0
oa = -c,a = -25c.
, a = -c, chon c = -l,a = l,b = 0 . Phuong
trinh
(a): x - z - 8 = 0
. a = -25c, chon c = -l,a = 25,b = 16 .
phuong
trinh
(a): 25x + 16y - z + 96 = 0.
3j
Vi (P) chua dj nen phuang
trirJi
c6 dang: ax + by + cz + 7a + b - 8c = 0
Voi a^ + b^ + c^ ^ 0 va 3a - 2b + c = 0 c = 2b - 3a
Theo

giathie'tbaitoan, taco:
d(I,(P))
=
^''" ^
42
9a-2b-7c
7a2+b2+c2
3
3 30a
-
16b
=
76(103^
-
12ab
+ 5h^) /
o 3(15a - 8hf = 2(103^ - 12ab +
5b^)»13403^
- 1428ab +
379b2
= 0
379,
1,
<»a = b, a = —b.
670 2
• a = -b, chon b = 2,a = l,c = l.
2
Phuang
trinh
mat phing (P) la: x + 2y + z +1 = 0 .

379,
• a =
•
-h, chon b = 670,3 = 379,c = 203.
670 ,
Phuong
trinh
(p) la: 379x + 670y +
203z+
1699 = 0.
Bai
6,2.3. Trong khong gian voi hf true toa dp
Oxyz,
cho 2 diem A(2;0;l),
B(0; -2; 3) va mat phSng (P): 2x - y - z + 4 = 0. Tim toa do diem M thuoc (P)
saocho
MA = MB = 3.
(Trichcau6adethiDHKhdiA-20n)
Jiie&ng
dan
gidi
Gpi E la
trung
diem AB ta c6: E(l;-1;2), AB =
(-2;-2;2)
Phuong
trinh
mat phing
trung
true (Q) ciia AB c6 phuang

trinh:
x
+ y-z + 2 = 0.
Vi
MA = MB nen suy ra M e (Q) => M e (P) n (Q)
Gpi
M(a;b;c)
suy ra:
[2a-b-c + 4 = 0
a+b-c+2=0
Mat khac: MA^ = 9 (a - 2)^ +
ia
+ 1
2
c=3+-a
2
b=l+-a
2
+ (|a +
2)2=9.
241
'IWd^^
phap
gtat
loan
titnh hoc
tfieo
chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu
6
Giai

ra ta
duoc
a = 0,a = -
•
hai
diem thoa yeu cau bai toan la: M(0;1;3), M
6 4 12^
'7'7' 7
Bai
7.2.3.
Trong khong gian vdi h$ tryc tpa dp Oxyz, cho mat cau (S)
phuong
trinh
x^ + y^ + z'^ - 4x - 4y - 4z = 0 va diem A(4;4;0). Viet phuorip
, 6
trinh
mat phang (OAB), biet B thupc (S) va tam
giac
OAB deu.
(Trich cau 6b
dethi
DH
Khoi
A - 201])
Jiuang
ddn
gidi
Xet
B(a;b;c).
Vi tam

giac
AOB deu nen ta c6 h^:
OA
= OB
OA
= AB
a^+b^+c^=32
(a-4)^+(b-4)^+c2=32
a+b-4=0
a=4-b
(^=31-a^-h^
c2=16-2b2+8b
Ma B e (S) nen : a^ +
b^
+ c^ - 4a - 4b - 4c = 0
<:>(4-b)^+b2
+]6-2b2+8b-4(4-b)-4b-4c
= 0
Hay c =
4=>b2-4b =0r^b
= 0,b =
4.Dod6
B(4;0;4)
hoac
B(0;4;4).
B(0;4;4)
taco
=
(16;-16;16)
OA,

OB
nen phuong
trinh
(OA/3): x - y + z = 0.
•
B(4;0;4)
taco
rOA,OB
1
=
(l6;-16;-16)
nen phuong
trinh
(OAB): x - y - z = 0 .
Bdi
8.2.3.
Trong khong gian he tpa dp Oxyz, cho duong thang
X
—
2
V
+
1
z
A:
—j—
= ^—^ = — va mat phang (P):x + y + z- 3 = 0.
Gpi
I la giao diern cua A va (?). Tim tpa dp diem M thupc (P) sao cho
MI

vuong goc vai A va MI = 4\/l4 .
Jlitang
ddn
giai
Taco
A cat (P) tai
I(l;,l;l).
Diem
M(x;y;3 -
x-y)e(P)
=^ MI = (l -x;l - y;x + y-2)
Duong
thing
A c6 a = (l;-2;-l) laVTCP
Ta CO:
MI.a
= 0
y-2x-l
MI^
=16.14
[(1 - x)^ + (1 -
y)^
+ (-2 + x +
y)^
= 16.14
242
Cty
TNHH
MTV
UVVII

Khang Vi^t
<=>
i
hoac
\y = -7
x = 5
y
= 9
Vay
CO
hai diem thoa yeu cau bai toan:
M(-3;-7;13)
va M(5;9;-ll).
,i
^ x-1 y z+2
gai
9-2-3.
Trong khong gian tpa dp Oxyz, cho duong thang A: — = -
-1
va m?t phSng (P): x - 2y + z - 0. Gpi Cla giao diem cua A vdi (P), M la diem
thupc A . Tinh khoang
each
tu M den (P), biet MC = V6 .
Jiu&ng
ddn
gidi
x = l + 2t
Cdch
jf:
Phuong

trinh
tham so ciia A:
Y
= t ,t€R.
z = -2-t
Thay x, y, z vao phuong
trinh
(P) ta dupe :
l
+ 2t-2t-t-2 = 0<»t =
-l=>C(-l;-l;-l).
'
Diem
Me A<»M(l + 2t;t;-2-t)=i>MC = >/6 o(2t +
2)2+(t
+ l)2+(t + l)2 =6
t
= 0
M(l;
0;-2) ^
d
(M;
(P)) =
t
= -2 ^
M(-3;
-2; 0)=^d
(M;
(P))
=

^
Cdc/i
2;
Duong
thang A c6 u = (2;l;-l) laVTCP
||< Mat phang (P) c6 n =
(1;-2;1)
la VTPT
cos^u,nj
'
Gpi
H la
hinh
chieu cua M len (P), suy ra cos
HMC
=
ip^entaco
d(M,(P)) = MH = MC.cosHMC = -^.
Bai
10.2.3.
Trong khong gian tpa dp Oxyz, cho cac diemA(1;0;0),
B(0;b;0)
C(0; 0; c), trong do b, c duong va mat phang (P): y - z +1 = 0 . Xac
dinh
b va
c, biet mat ph^ng (ABC) vuong goc vdi mat phang (P) va khoang
each
tu
gdiem
O den mat phMng (ABCD) bang , ,

^
, ,
(Trich cau 6a
dethi
DH
Khoi
B -
2010).
M
Jiuang
ddn gim u
M
Phuong
trinh
(ABC): - +1 + - =
1
;K^'I
of .
^
,! 1 b c
243
Phucnig
phcip
giai
Todn Hiith hpc
theo
chuyen
de-
Nguyen Phu Khdnh, Nguyen
Tat

Thu
Vi
(ABC)
±(P)=i>
=
0<»b
= c=i>
(ABC)
:bx +
y
+
z-
b =
0.
b
c
Ma
d(0,(ABC))
=
-
=
i
<=>
b =
- (do
b > 0 ).
Vay
b = c =
—
la gia tri can tim.

k '
Bdi
11.2.3.
Trong
khong gian toa
dp
Oxyz,
cho hai mat phang (P):x+y+z-3-{
va
(Q):x-y
+ z- l = 0.
Viet phuang
trinh
mat phang (R) vuong
goc
voi (P)
va
(Q) sao cho khoang each
tu O den
(R) bang
2.
(Trich
cm
6a dethi DH Khoi
D -
2010
)
Jiucmg
dan giai
=

(1;0;-1)
laVTPT
Mat
phang (P)
c6 np
=(1;1;1) la
VTPT,
mp(Q)
c6 ng
= (1;-1;1) la
VTPT.
^J(R)i(P)^
ir-
l(R)i(Q)
Suy
ra
(R):
x
-
z + m =
0
m
mp(R)
CO
n^
=-|^np,nQ
Taco
d(0;(R))-2<=>
•
=

2
<=>
m
=
±2N/2
Vl
+
0 + 1
Vay
(R):x-z±2V2
=0.
Bdi
12.2.3.
Trong
khong gian
voi
h§
toa dp
Oxyz,
cho tu
dien
ABCD
c6
cac
dinh
A(1;2;1), B(-2;1;3),
C(2;-l;l)
va
D(0;3;1).
Viet phuang

trinh
mat
phang
(P) di
qua A,
B
sao
cho khoang each tir
C den (P)
bang khoang each
tu
D den
(P).
(Trich
cm
6a dethi DH Khoi B
-
2009
;
Jiudng
ddn gidi
Mat
phang (P) thoa man yeu cau bai toan trong hai truang hpp sau:
Trucmghp'pl:
(P) di
qua A,B song song voi CD.
Ta
CO AB =
(-3;-l;2),
CD =

(-2;4;0), suy ra n
=
rAB,CD]
= (-8;-4;-14)
la
VTPT
eua (P). Phuong
trinh
(P):
4x + 2y + 7z -15
- 0.
Truang
hgp 2:
(P) di
qua A,B
va
cat CD
tai
I, suy ra
I la
trung diem cua
CD
Dodo I(1;1;1)=>AI = (0;-1;0).
Vec
to
phap tuyen eua mat phang
(P):
n =
AB,
All

=
(2;
0;
3).
Phuong
trinh
(P):
2x + 3z
-
5 =
0.
244
Va)
Cty
TNHH
MTV
DWH
Khang Viet
Vay
(P):
4x + 2y + 7z -15 =
0
hoac
(P):
2x + 3z
- 5
=
0. , i <
gCii
13.2,3.

Trong
khong gian
voi he toa dp
Oxyz
cho
A(2;5;3)
va
duong
d:-
•1
_ y _ z-2
.
Tim
toa dp
hinh chieu vuong
goc
cua
A len d va
2
1 2
viet phuong
trinh
mat phing
(P)
chua duong thSng
d sao
cho khoang each tir
den
(P)
Ion nhat.

[Trich
dethi DH khoi
A -
2008
)
Jiuorng
ddn giai
,
Duong thang
d c6 uj
= (2;1;2) la
VTCP.
Gpi
H
la hinh chieu
ciia
A
len
d
r:^
H(1 +
2t;
t;2 + 2t)
^
AH = (2t -1;
t
-
5;2t -1)
Do AHld=>
AH.u7

=
0«2(2t-l)
+
t-5
+ 2(2t-l) =
0
<*t
=
l=>H(3;l;4).
•
Gpi H' la hinh chieu cua
A
len mp(P).
Khi
do,
ta
eo:
AH'
<
AH => d(A, (P)) Ion nhat
<=>
H
= H'
o
(P)
1
AH
Suy
ra AH
=

(1;
-4; 1) la
VTPT
cua (P) va (P) di qua H.
^ -
Vay
phuong
trinh
(P): x-4y +
z- 3
=
0.
Bdi
14.2.3.
Trong
khong gian
voi h? tpa dp
Oxyz,
cho ba
diem A(0;l;2)
B(2;-2;1),C(-2;0;1).
1) Viet phuong
trinh
mat phang
di
qua ba diem A, B,
C
va tim tpa
dp
true tam

tamgiacABC.
2)
Tim
tpa dp
cua diem
M
thupc mat phang
(P):
2x
+ 2y + z - 3 = 0 sao cho
MA
=
MB
=
MC.
JJu&ng
ddn gidi
l)Tac6:
AB =
(2;-3;-l),AC
= (-2;-l;-l) =>
rAB,Ac]
= (2;4;-8)
la mot
VTPT
ciia
mp(ABC).
Phuong
trinh
mp(ABC):x

+ 2y-4z + 6 =
0.
Gpi
H(a;b;c)
la
tryc
tam tam giac ABC
•
H €
(ABC)
^a
+2b-4c+
6
=
0
Taco:
CH
= (a;b-l;c-2),
B"H
= (a-2;b + 2;c-l)
CH1
AB
BHIAC
Vi
AB.CH
=
0
BH.AC
=
0

Tir(l)
va
(2) suy ra
a
= 0;b = l;c =
2.
Vay
H(0;l;2).
2a-3b-c +
5
=
0
2a+b+c-3=0
(2)
Phuong
phdp
gidi
ToAn Hinh hoc
theo
chuyen
de-
Nguyen
Phu Khdnh,
Nguyen
Tai Thu
2) Gia sir M(a;b;c)6(P)=>2a + 2b + c-3 = 0
-2b - 4c + 5 = -4a + 4b - 2c
+
9
Do

MB^
= MC^
<=>
-4a +
4b-2c
+
9-4a-2c
+ 5
<=>
2a - 3b - c = 2
2a-b = l
(3)
(4).
Tir
(3) va (4) ta tim duoc: a = 2; b = 3; c = -7
Vay
M(2;3;-7)
la diem can tim.
Bdi
15.2.3.
Tim m,n de 3 mat phang sau ciing di qua mgt duong thang:
(P):x + my + nz-2 = 0, (Q)
:
x + y - 3z +1 = 0 va (R): 2x + 3y + z-l = 0.
Khi
do hay viet phuong
trinh
mat phang (a) di qua duong thang chung do
23
va tao voi (P) mot goc cp sao cho

coscp
= . .
\/679
Jiuang
ddn
gidi
x+y-3z+l=0
2x + 3y + z-l = 0
Cho z =
1
=> X = 6, y = -4 A(6; -4; 1) € (Q) n (R).
Cho z = 0 X = -4, y = 3
=>
B(-4; 3; 0) e (Q) n (R).
Ba mat phang da cho cung di qua mgt duong thang
<=>
A, B e (P)
Xet h$ phuong
trinh:
-4m + n = -4
3m = 6
m-2
n
= 4
la gia
tri
can tim.
Taco:
n-(l;2;4)
la

VTPTcua
(P)
Vi
(a) di qua A nen phuong
trinh
ciia (a) c6 dang:
a(x-6)
+ b(y + 4) + c(z-l) = 0
DoB 6 (a) nen
taco:
c =-10a + 7b . Suy ra
v==(a;b;-10a+7b)
la
VTPTcua
(a)
Nen
theo
gia thiet ta c6:
cos9
=
n.v
-39a + 30b
—
n
•
v
2].Va^+b^+(7b-10ar
23
Suy ra
coscp

= . o
-39a+30b
23
V679
V^.^a2+b2+(7b-10a)2 ^679
•
>/97|39a
-30b| =
23^3(l01a2+50b2-140ab
•
3.97(l3a
-lOb)^ = 23^ (lOla^ -
140ab
+ 50b^ j
^ 85a^ +
32ab
- 53h^ = 0 a = -b,a = —b
85
Cty TNHH MTV DWH Khang Viet
^ a = -b ta chon b = -1 => a = l,c = -17 . ^i, .^/:
phuong
trinh
(a):
x-y-17z
+ 7 = 0 .
, a = —b tachon b =
85=>a
= 53,c = 65. ^ '
85
phuong

trinh
(a): 53x + 85y + 65z - 43 = 0 .
gcii
16.2,3,
Lap phuong
trinh
mat phang (a) di qua diem M(l; 9; 4) va cat cac
tfuc toa do tai cac diem A, B, C
(khac
goc tpa dp) sao cho
J) M la true tarn cua tarn
giac
ABC ' • '« !
2) Khoang
each
tu goc toa dp O den mat phSng (a) la Ion nha't. •' ' '
3) OA = OB = OC d g.Tu;i/i'-;
4) 80A = 120B +16 =
370C
va x^
>0,zc
<0. KJ\'> rtvv;*
Jiuang
ddn
gidi
' ' '
Gia sir mat phang (a) cat cac true tpa dp tai cac diem
khac
goc toa dp la
A(a;O;0),B(O;b;0),C(O;0;c)

voi a,b,C7iO. ,
Phuong
trinh
mat phang (a) c6 dang
—
+
—
+
—
= 1. •' '
a b c ,
»•.'.
19 4
Mat
phang (a) di qua diem M(l;9;4) nen - +
—
+ (1).
a b c
1) Ta c6:
AM(1
- a; 9; 4), BC(0; - b; c),
BM(1;
9 - b; 4), CA(a; 0; - c).
Me (a)
Diem
M la true tam tarn
giac
ABC khi va chi khi
AM.BC
= 0

BM.CA = 0
I
19 4,
-+—+-=1
a b c
9b = 4c
a = 4c
Qc
u 98 49
•a = 98;b = —;c=: —.
9 2
huong
trinh
mat phang (a) can tim la x + 9y + 4z - 98 = 0.
\Cdchl:Tac6:
d(0,(a))
=
-1
J A. JL J_ i_ JL
a^'^b^^c^
ia'\^\
It
h.'l
Bai toan tro thanh, tim gia trj nho nha't cua T = + ^ + ^ voi cac so' thuc
a^ •
19 4
•,C7t0
thoa man-
+
-

+
- =
1
(1).
a b c
Phuontg
phap
giai
Todn
Hinh
hoc
theo
chuyen
tie
-
Nguyen
Phil
Khdnh,
Nguyen
Tat Thu
Ap
dung bdt Bunhiacopski
ta c6:
x2
1.1.9.1.4.1
a
b c
1
<(l2+92+42)
'1 J_ 1'

Nen suy
ra T > — .
Dau dang thiic xay
ra
khi
1:1
=
9:1
=
4:1
a
b
19
4,
a
b c
*^
c^a =
9b
=
4c
=
98.
Phuang
trinh
mat phang
(a)
can tim
la x +
9y

+
4z
-
98
= 0.
Cdch 2;Goi
H la
hinh chieu ciia
O
tren mat phling
(a).
Vi
mat phang
(a)
luon di qua diem co'dinh
M nen
d(0, (a))
=
OH
<
OM
=
^98.
Dau dang thuc xay
ra
khi
H =
M, khi do
(a) la
mat phang di qua

M
va
c6
vec to phap tuyen
la
OM(l;9;4) nen phuong
trinh
(a) la
1
.(x -1)
+
9(y
-
9)
+
4.(z
-
4)
= 0 o
X
+
9y
+
4z
-
98
=
0.
, do do xay ra bon truong hop sau:
3)

Vi
OA
=
OB
=
OC
nen
• Truong hp'p 1:
a = b = c.
19
4
Tir
(1)
suy ra
—
+
—
+
—
=
1 <» a
=
14,
a
a a
nen phuong
trinh
(a) la: x + y + z -14 = 0.
19
4

• Truong hop
2: a = b = -c. Tu (1)
suy
ra - + =
l<=>a
= 6,
nen phuong
a
a a
trinh
(a) la x + y - z - 6 = 0.
19
4
• Truong hop
3: a = -b = c. Tu (1)
suy
ra + - = loa =
-4, nen phuong
a
a a
trinh
(a) la x - y + z + 4 = 0.
19
4
• Truong hg-p
4: a = -b = -c. Tu (1) c6 =
1
o a =
-12,
nen

phuong
a
a a
trinh
(a) la x - y - z +12 = 0.
V^y
CO bon mat phang thoa man la
x + y +
z
-14 = 0,
va
cac
mat phing
x
+
y-
z-
6
=
0,x-y
+
z
+
4
=
0, x-y-z
+
12
= 0.
4)

Vi x^ > Q,ZQ < 0
nen
a > 0,c <
0, do do
80A
=
120B
+16 =
370C
8a =
12|b|
+16 =
-37c.
Cty TNHH MTV
DWH
Khang
Viet
, Neu
b>0=^c
=
-^a,b
= ^,a>2
nen tu
(1) ta c6
1
27 37
a^2a-4
2a
= 1 <=>
+

2a
-
35
=
0
»
a
= 5
a
= -7
Vi
a > 2
nen
a = 5 r> b = 2;c = -1^,
phuong
trinh
mat phang can tim
la
(a):8x
+
20y-37z-40
= 0.
8
,
4
-
2a
. Neu
b<0=>c
=

-—a,b
= -^
,a>2 nentu
(1) ta c6
1
27
—
+
•
37
= 1
o
a^
+
29a
-
35
=
0
a ='
-29
+
3N/T09 r;|^ :
a
4-2a 2a ^ ^
-M-IHA: '
^-qojAl
.
Vi
a > 2

nen khong
c6
gia tri thoa man.
y
Vay phuong
trinh
mat
phSng
(a): 8x
+
20y
-
37z
-
40
= 0.
Bdi
17.2.3.
Trong khong gian Oxyz cho duong
thSng
dm
c6
phuong
trinh
x-3m-l
y-l__
z-6m
,
m
i

B
2m
1
- m
3m
+1
B
Chung minh rang khi
m
thay doi, duong thang
dm
luon nam trong mQt mat
phang
(a) co
dinh.
Viet phuong
trinh
mat phang do.
Jiuang
dan gidi
t
Duong thang
dm
di qua A(3m
+
l;l;6m) va
c6
VTCP
u =
(2m;

1
-
m;l
+
3m)
Xet mat phang
co
djnh (a): ax
+
by
+
cx
+
d
= 0
chua
dm.
Khi
do:
Ae(a)
l
,Vm
n„.u
= 0
^ a(3m
+
l)
+ b +
c.6m
+

d
= 0 ^
[a.2m
+
b(l-m)
+
c(l
+
3m)
= 0' ^
(3a
+
6c)m
+
a
+
b
+
d
= 0
(2a
-
b
+
3c)m
+ b + c = 0'
a
= -2c
b
= -c ,

chpn
c =
-1
a = 2,b =
l,d
= -3
d
= 3c
3a
+ 6c = 0
a+b+d=0
2a-b
+
3c
= 0
b
+ c = 0
Vay
dm
luon nam trong mat phang (a):
2x + y - z - 3 - 0.
949
Phuatig
phdp
gidi
Totin Hinh hoc
theo
chuyen
de -
Nguyen

Phu Khdnh,
Nguyen
Tat Thu
Bdi
18.2.3.
Trong khong gian Oxyz cho tu
di§n
ABCD c6
A(l;
1;
1), B(2; 0; 2), C(-l;
-1;
0), D(0; 3; 4)
AB AC AD
Tren cac cgnh AB, AC, AD lay cac diem B', C', D' thoa + + = 4
^ AB' AC AD- •
1) Viet phuong
trinh
mat
phSng
(B'C'D')
biet tu di^n AB'C'D' c6 the tich Ion nha't,
2)
Viet phuong
trinh
mat phJing (a)
song song
voi AB, CD cat AC, BD Ian luot
tai
M va N thoa AM + BN = 6.

Jiuorng
dan
gidi
1) Ap dung bdt Co si ta c6:
AD
.J AB.AC.AD AB'.AC'.AD' 27
- => > —
64
, AB AC
4 = + +
>33,
AB'
AC AD' VAB'.ACAD'
Suy ra
VAB'CD'
_ AB'.ACAD'^ 27
AB.ACAD
~64
AB.AC.AD
27
V
ABCD
•
^AB'C'D'
-
64
^ABCD
Do do V
AB'C'D
• nho nha't khi va chi khi

• AB'^AC^AD' 3
AB AC AD ~ 4
Hay mat phang
(B'CD')//(BCD).
BC = (-3;
-1;
-2), BD = (-2; 3; 2) :^ BC A BD - (4; 10; -11)
AB'
= -AB:
4
yB'-i==-|-B'
7 !_ 7
4'4'4
V|y
phuong
trinh
mat phang
(B'C'D')
la: 4x + 10y-llz + —= 0
.i
4 ;
2) Tir gia thiet, ap dung djnh li
Talet
ta c6: '
AN
BM
AC
BD
•BM = 2AN=>AN-2
Suy ra AN = -ACo

3
-1 = -1
, 2
Ma n„ = AB A CD = (-8; -3; 5)
Phuong
trinh
(a): 8x + 3y - 5z + — = 0.
3
Cty TNHH
MTV
DWH
Khang
Vi$t
I
§ 3. PHirONG TRiNH
DLTONG
THANG
Hfoe lap phuong
trinh
duong
thSng
A, ta can tim mgt diem M(xo;yo;zo) ma
^ di qua va mQt
VTCP
u^(a;b;c).
Khi do phuong
trinh
cua A c6 dang: :
'x =
Xg

+at '
• •
•••••>-;••'
•
y = yQ + bt, te M .
•i:.x
<uc
Z
= ZQ + ct
Khi
tim
VTCP
cua A, chiing ta can luu y:
•
dr/v; ;>rr»;;fv
r
, Neu A di qua hai diem A, B thi AB la
VTCP
cua A.
, Neu A 1 (P) thi
il^
- . • -^''"^^
. Neu A//d thi u^ = u^. ai^ cJ'
• Neu hai vec to khong cung phuong a,b c6 gia vuong goc vai A thi vec to
ji = a A b la mpt
VTCP
cua A.
. Neu
Ac:(P)
thi u^ Inp .

• Neu A la
giao
tuyen cua hai mat phang (a) va (P) thi u =
n^^
A np la mpt
VTCP
cua A. Trong do n^,n^ Ian lugt la
VTPT
ciia (a) va (P).
Khi
chpn mat phang
chua
A, chiing ta can luu y:
• Neu A di qua M va vuong goc voi duong thang d thi A nam trong m^t
phang (P) di qua M va vuong goc vdi d.
• Neu A di qua M va cat duong thMng d thi A nam trong mat phang (P) di
qua M va d.
• Neu A di qua M va
song song
voi mat phang (a) thi A nam trong m^t
phSng
(P) di qua M va
song song
voi mat phang (a).
_ 1
Vi
du
1.3.3.
Lap phuong
trinh

duong th5ng A biet: ^ ,
1) Adiqua A(1;2;1) va
B(-1;0;0)
2) A la
giao
tuyen cua hai mat phMng:
(a):x+y-z+3=0
va
(P):2x-y+5z-4
= 0
^ Adiqua M(1;0;-1) va vuong goc voi hai duong thang
. x y+2 z-1 . . .
. 0- ,u. •
x =
t••••'•'••-•'^
". •
y
= -l-2t.
z = 0
h) Vi A di qua A va B nen A nhan AB =
(-2;-2;-l)
la
VTCP
Phicamg
phdpgiai
Todn Hiith UQC
theo
chuyen
de -
Nguyen

Phii Khdtih,
Nguyen
Tat Thu
x = l-2t
Phirong
trinh
tham so' cua A:<y = 2-2t,teM.
z = l-t Ajv.
2) De lap phuang
trinh
duong thang A ta c6 cac
each
sau
Cdch
l:Ta c6
M(-l;-l;l),
N(-5;6;4)
la hai diem chung ciia (a) va (P)
Suy ra
M,
N € A
=>
MN = (-4; 7; 3) la mpt VTCP ciia A.
x = -l-4t
Phuong
trinh
tham so' cua A: y = -1 + 7t, t e
M
.
z = l + 3t

Cdch
2;Ta c6 c6 ri;; -
(1;1;-1),
np =
(2;-l;5)
Ian
lugt
la VTPT ciia (a) va (P)
Vi
A la giao tuyen cua (a) va (P) nen u = n^
A
np' =
(4;-7;-3)
la VTPT cua A.
fx
=
-l
+ 4t
Va diem M e A nen phuong
trinh
A:
•
y
= -l-7t, teM.
z = l-3t
Cdch
3: Ta c6 M(x; y; z) e
A
o ^ o
Me(p)

x+y-z+3=0
2x-y + 5z-4 = 0
Datz = ttadu<?c:
j''"'^"
[2x-y = 4-5t^
1
4
x = 1
3 3
10 7
y
=
+
_t
^ 3 3,,
J§f Phuong
trinh
tham so ciia A: j
1
4,
x = 1
3 3
10 7
y y +
-t, t€K.
z = t
3)
Tac6:dic6
u
j'=

(5;-8;-3)
laVTCP;d2c6 u^-(l;-2;0) la VTCP
Cdch
7;Gia
sir u =
(a;b;c)
la mot VTCP ciia A.
Vi
A vuong goc voi di va d2 nen ta c6:
fu.ui=0
f5a-8b-3c
= 0 f^"^^ _ h
Phuong
trinh
A la:
c = -b
3
X
=
1
+ 6t
y
= 3t ,teK.
z =
-l
+ 2t
CtyTNHll
MTV DVVH KIniiig
Vict
2: Vi A vuong goc vol hai duong thang

di,
d2nen vec to
u
=
Uj
A
U2
=
(-6;-3;-2)
la VTPT ciia duong thang A.
x = l-6t ;;:tj;vlb'-
z = -l-2t
Vay phuong
trinh A
:
X^i
du
2.3.3.
Lap phuong
trinh
duong thSng A, biet
1) A nam trong (P): y + 2z = 0 va cat hai duong thang
x = l-t
x = 2-t' -V'^«:i-j,
jv-,:,,
di:.
y
= t ; dj-
y=4+2t'. ,
z = 4t

z = l
2) A di qua
M(-4;-5;3)
va cat hai duong thang
x+1
y+3 z-2
di
. = —
1
3 -2-1
, , x-2 y+1 z-1
va d, : = =
2 3-5
3) A di qua M(0;1;1), vuong goc voi di' —-= ^^'^ =— va cat duong
x =
-l
thang d2 :
-y
= t .
z = l + t
J:ffigidi.
1) Goi A, B Ian luot la giao diem ciia (P) voi
di,
d2
Thay phuong
trinh
dj vao phuong
trinh
(P) ta c6 :
t

+ 8t = 0<=>t = 0=> A(1;0;0).
Thay phuong
trinh
d2 vao phuang
trinh
(P) ta c6 :
2t + 6 = 0 t =-3
=>
B(6;-2;l).
Vi
A nam trong (P) dong
thoi
A cat
di,
d2 nen A di qua A, B
SuyraA nhan AB =
(5;-2;1)
lam VTCP.
Vay phuong
trinh
ciia duong thang A la:
x = l + 5t
y
= -2t , tG
z = t
2) Ta
CO
didi
qua Mi(-l;-3;2) va c6 u^
=(3;-2;-l)

la VTCP
Duong
thang d2di qua M2(2;-l;l) va u^ =
(2;3;-5)
la VTCP
A.
Gpi
(P) la mp di qua M va chiia duong thang di.
Khi
do (P)
CO
np = MMj
A
Uj
=
(-4;0;-12)
la VTPT
253
Pliinri!};
pluip
i^iai
Todii lltiih
hoc
llu-o
chuyen rfe - Nguyen Phu
Khdnh,
Nguyen Tat Thu
Tuong
tu
gpi (Q)

la mp
di qua M
va
duong thing d2,
suy
ra
=
(7;-13;-5) la
VTPT
cua (Q).
Vi
A
di qua M va
cat
hai
duong thing
di,
d2nen
A
la
giao
tuyen cua (P)
va
(Q)
Suy
ra
u^
=
n^
A

n^ =
-52(3; 2;-1)
la
VTCP
cua
A.
x
=
-4
+
3t
Phuong trinh
A
: •
y =-5
+
2t, t e
K
.
z
=
3-t
3)
Cdch
J;Du6ng thang d2
di
qua
A(-1;0;1) va
c6
u^ =

(0;1;1) la
VTCP
Duong
thing di
c6
u^
=
(3;1;1)
la
VTCP
Gpi
(P) la mp di qua M
vuong
goc
vol duong thang di,
suy
ra
np
=
(3;l;l)
la
VTl^
ciia
mat
phing
(P).
Goi
(Q)
la mp di
qua

M va cat
duong thing d2. Suy
ra
rig
=
U2
AMA=(1;-1;1)
la
VTPT
cua mat
phang (Q).
Vi
A
la
giao
tuyen
cua
hai
mat
phang
(P) va
(Q) nen suy
ra
uX
=
%
A
ri^
= 2(1; -1;
-2)

la
VTCP
cua A.
x
=
t
y-l-t
,teR.
Phuong trinh
A:
z
=
l-2t
Cdch
2:Gpi
N
=
A
n
d2
N(-l;
t;
t
+1)
MN =
(-1;
t
-1;
t)
Vi

A
vuong
goc
voi
dj
=>
MN-u^
=
0
<=>
t
=
2
=> MN =
(-1;
1;
2)
x t
y=l+t
,teR.
z
=
l
+
2t
Phuong trinh
A:
Vi
du
3.3.3.

Trong khong gian Oxyz
cho
duong thang
A:
x
=
l
+
t
y =
2
-1
va mat
z
=
l
+
2t
phang
(a):2x
+
y
+ 2z-ll =
0.
1)
Lap phuong trinh hinh chieu cvia
A
len
mat
phang

(a)
2)
Viet phuong trinh duong thing
Ai
nam
trong
(a)
dong thoi
cat
va
vuong
goc
voi A.
3)
Viet phuong trinh duong thing
Aj
nim
trong
mat
phing
(a), cat A va
*
7^3
tao
voi duong thing
A
mpt
goc
cp
thoa

coscp
=
.
18
254
Cty
TNHHMTV
I>VVll Khang Viet
Xffigidi.
Duong
thing
A
di qua M(l;2;l)
c6
VTCP
u^ =
(1;-1;2)
Mat phang
(a) c6
VTPT
n„
=(2;1;2).
x
=
l
+
t
J)
Cdch l;Ta
thay

h§
phuong trinh
y =
2-t
z
=
l
+
2t
2x
+
y
+ 2z-ll =
0
x
=
2
y =
l
z
=
3
|p,Suy
ra
An(P) = I(2;l;3).
Goi
H la
hinh chieu ciia
M
len

(a),
suy
ra
phuong trinh MH:
Suy
ra
H(l
+
2t;2
+
t;l
+
2t).
x
=
l
+
2t
y = 2
+1
z
=
l
+
2t
Do
He (a)
nen
ta c6 :
2(l

+
2t)
+
2
+
t
+
2(l
+ 2t)-ll
= 0<=>t
=
^ i „r ,
Hay
H
.IH
=
'19
23 19'
9'9'9/
Hinh
chieu
cua
A len
mat
phang
(a)
chinh
la IH
1
14 _8

9'9'
9
_
x-2 _
y-1_
z-3
Phuong trinh IH:
^
1 14 -8
Cdch
2.
Goi
(P)
la mat
phing chua
A va
vuong
goc
voi
(a)
Suy
ra np =
n^,
A
u^
=
(4;-2;-3).
Gpi
A'
la

hinh chieu
cua A
len
mat
phang
(a),tac6
A' =
(a)n(P)
Suy
ra
u^.
=
n^
A
np =
(l;14;-8)
va
I
e
A' nen phuong trinh
A' la
:
x-2_y-l_z-3
1
" 14 " -8
2)Gpi
J =
AinA=>J
=
An(a)i:i>J =

I
Ain(a)
Vi
Aj
lA
"AI
=n„AU^=(4;-2;-3)
.x-2_y-l_z-3
Alt.
!
I
Phuong trinh
Aj:
^ ^ ^ .
3)
Vi
Aj
nam
trong
mat
phang
(a)
va cat A nen Aj
di qua
I.
Gpi
u.
=(a;b;c),
taco
:

UA^.na
=0o2a
+ b +
2c
=
0ob
= -2a-2c
255
Phuongphdpgiai
Toan Hinh hpc
theo
chuyen
de -
Nguyen
PhuTiluhih,
Nguyen
lat inW
Mat
khac
: cos
9 =
7
UA2
UA
a-b + 2c|
UA
3a
+ 4c'
V6(a2+b2+c2)
76(5a2

+8ac+5c2)
Do do cos
(p
= —— o
3a + 4c
1^
^6{5a^
+8ac
+
5c2)
18
3^2
3a + 4c
=
zVsa^
+8ac
+
5c2
o 18(3a + 4cf =
49(53^
+
8ac
+ Sc^)
<=>
83a^ -
40ac
- 43c^ = 0
<=>
a = c,a = -—c
83

• a = c => b = -4a, chon a=l=>b = -4,c = l
ou
.^ u A y-1 z-3
Phuong
trinh A2
:
—-—
=
——
=
—.
43
• a =
cchon
a =
43=>b
= 80,c = -83
83
Phuong
trinh A2
:
x-2_y-l_z-3
43 80 -83
Vidu
4.3.3.
Trong khong gian Oxyz cho hai duong thang
di
:•
x-1 y+1 z , , x-3 y+3 z-2
• 1 = — va do : = =

1
2-2 ^2 1-1
1) Viet phuong
trinh
duong thSng A cat hai duong thang di,
d2
Ian
lugt
tai
M,
N thoa MN = 3\f3 va MN
song song
voi mp (a): x - 2y + 3z = 0 .
2) Viet phuong
trinh
duong thang d di qua O va cat dj tai A sao cho mat
cau tarn A va di qua O tiep xiic voi mat phang (P): 5x + y - 4z -16 = 0.
JCffigidi.
1) Vi MedpNedj
nensuy
ra M(l + m;-l+ 2m;-2m), N(3 + 2n;-3 +
n;2-n)
Do do MN = (2n-m +
2;n-2m-2;-n
+ 2m + 2)
Theo
de bai ta c6: '
MN^
= 27
<=> <

(2n - m + 2f + 2(n - 2m - if = 27
MN.na
=0 2n-m +
2-2(n-2m-2)
+ 3(-n + 2m + 2) = 0
o
n
= 3m + 4
(5m+10)2 +2(m +
2)2
=27
m
=
-l
n
= l
m
= -3
n
= -5
m
=
-l
n
= l
, suy ra M(0;-3;2),MN =
(5;-2;l)
Phuong
trinh
A :

—
= =
^ 5-2 1
256
Cty TNHH MTV DV\'H Khang Viet
m
= -3
n
= -5
, suy ra M(-2;-7;6),MN =
(-5;-l;l)
X + 2 -_ y + 7 _ z - 6
I phuang
trinh
A: ^ ^ ^ ,
2)
Taco
Aedj nen
A(3
+ 2a;-3 + a;2 - a)
Vi
mat cau tarn A di qua O tiep xuc voi ((3) nen ta c6 : OA = d(A,(P))
Ma OA^
=(3
+ 2a)2+(a-3)2+(a-2)2 =6a2+2a + 22
d(A,(P))
=
15a-12
V42
nen ta co :

v2
6a2
+
2a
+
22
=
^1^^^-^
o
9a2
+
148a
+
260
=
0
s=>
a =-2,a =-—
42 9
• a =
-2,suy
ra A(-l;-5;4) => u^ = OA =
(-l;-5;4)
X y z
Phuong
trinh
d:
—
=
—

= — .
1
5 -4
130
• a = , suy ra A
Phuang
trinh
d:
^ 233 157 148^
9
' 9 ' 9
y
_ z
•
uj - -^90A = (233;
157;-148)
233 157 -148
Vi du
5.3.3.
Trong khong gian Oxyz cho hai duong thang
x-2_y + l_z x-3 _ y-2 _ z +
1
^-112^2-1-1
1) Chung
minh
rang hai duong thang A^^Aj
cheo
nhau. Viet phuong
trinh
duong vuong goc chung cua hai duang thang do.

2) Viet phuong
trinh
duong thSng A di qua O, cat Aj va vuong goc vai A2.
Xffigidi.
1) Ta de dang chiing
minh
dugcAj,A2
la hai duong thang
cheo
nhau.
Gpi
A la duang vuong goc chung cua hai duong thang Aj va Aj.
Taco
AnAj
= A(2-a; -1 + a; 2a),
AnAj
=B(3 + 2b; 2-b;
-1-b).
Do dompt vec tochi phuong ciia A la: AB(1 + 2b +
a;3-b-a;-l-b
-2a).
Vi
A1
A, ^
nen
<
ABlu^
AB.u^^
= 0
A1

A, ^
nen
<
'
<=><
AB.u^^
Al
A2
ABlu,^
AB.u^^
= 0
, do do
-l-2b-a
+
3-b-a-2-2b-2a
= 0 J4a + 5b = 0
2 + 4b + 2a-3 + b + a + l + b + 2a = 0 '^|5a + 6b = 0
Phuang
phap
gidi
Todn
Hinh
hoc
theo
chuyen
de -
Nguyen
Phu
Khdnh,
Nguyen

Tat Thu
Do do a = 0, b = 0 nen A(2; -1; 0), B(3; 2; -1), AB(1; 3; -1).
x = 2 + t
Phuang
trinh
duong vuong goc chung can tim la <
y
= -l + 3t
(teR).
z = -t
2) G(?i M = AnA^ =::>M(2-m;-l + m;2m)=>OM = (2-m;-l + m;2,m)
Vi
AIA2 nen MO.UA2 =
0<=>
2(2-m)-(-l + m)-2m = 0<» m = 1
Suy ra
UA
=OM =
(1;0;2).
x = t
Phuong
trinh
A :
<
y = 0 ,telR.
z = 2t
Vidu
6.3.3.Trong
khong gian voi h§ toa dp Oxyz cho hai duong thang
, x-3 y-3 z-3 , x + 5 y + 2 z

3 2
1) Chung
minh
rang dj va dj cat nhau tai I . Viet phuong
trinh
duong
thang A tao voi hai duong
thSng
di,
d2
mot tarn
giac
can tai I va c6 dien
tich
bang .
2) Viet phuong
trinh
duong phan
giac
cua goc tao boi hai duong thang dj
va dj.
Xffi
gidi.
1) Xet he phuang
trinh:
'x-3 y-3 z-3
2 2 1
x + 5 _ y + 2 _ z
6 ~ 3 "2
x = l

y
= l
z = 2
B-
Vay dj cat d2 tai
giao
diem
l(l;l;2).
di
di qua diem
Mi(3;3;3)
c6 ^ =
(2;2;1)
laVTCP;
dj
diqua M2(-5;-2;0)
vaco
u^-(6;3;2)
laVTCP.
Gpi
cp
la goc giiia hai duong thang dj va d2
Cty TNHH MTV DWH
Khang
Viet
a CO
:
cos
cp
=

U1.U2
.
"1
U2
20 . r — Vii
— => sm
(p
= vl - cos
(D
=
21 ^ ^ ^ 21
'a su lA = IB = a > 0 . Di?n tich cua tarn
giac
lAB la :
. 1 ^. ^„ .
2>/4T
ViT
S = IA.IB.sin(p = a = r:>a = l.
2 42 42
aco: Aedj => A(3
+2t;3
+ 2t;3 +1) => lA = (2t + 2;2t + 2;t +1)
Nen lA^
=lc^9(t
+
l)^
=lo
'^~^:^A,(^;^;^),A2(i;i;^).
t
= -l 3 3 3^ 2^3 3 3^

. " 3
B e d2 =^ B(-5 + 6t; -2 + 3t; 2t) =^ IB = (6t - 6; 3t - 3; 2t - 2)
o do: IB^
=lc:>49(t-l)^
=lc:>
t
= -
7
, 7
uy
ra Bj
13 10 16
7'7'7
1
1
11]
K7'7'7)
ay ta c6 4 truong hop la:
13 10 16^ '55.7
.3'3'3;
;B
;\3'3'3
;B
l7'7'7
5
4 -5
ri
4 12^ -V
—=>A
l7'7'7J

5 5
3-
I21'"
5
"21'
1
^
21.
z-
7
3
-1
32
23 13^
V
21'
"21'" "21.
z-
7
3
32 23
^1
1 5\0 16^
U
3 3.
\7 7 7
ri> AB =
13
^32 23 13>
21 21 21

1
1 5
X - - y - ~ z - -
Phuong
trinh
A : ^ ^
=-
^
32 23 13
'1.1.5>
;B
ri
4 i2>
=>AB
=
/
4.5.1'
.3'3'3>
;B
l7'7'7;
=>AB
=
\
"2l'2l'21.
Phuongphapgiai
Toau
Uhth hoc
thco
chuyen
de-

Nguyen
Phi'i
KhdnU,
Nguyen
Tai Thu
1
1 5
Phuong
trinh
A:
X
— V
— z-
3
_1 3 3
4
-5 -1
2)
Goi d la
duang phan
giac
cua goc tao boi hai duong thang d^dj
Ta CO led. Dat
=
1
f2.2.r
1
f6
3 2^
-Ui

=
.3'3'3y
112
-U2
=
[7'7'7)
Khi
do
ej,e2 la cac
vec
to
don vi tren d],d2 .
Suy ra = ej ±62
"d
=ei+e2
=
"d
=^1-62 =
32
23 13^
21 21 21
21'21'21
, suy
ra
phuong
trinh
d 1
-
x-1 _
y-1_z-2

^'
32 " 23 ~ 13
X
—
1
y""-! 2-2
, suy
ra
phuong
trinh
d la: = =
4
5 1
Vi
da 7.3.3.
Trong khong gian Oxyz
cho
hinh
chop
S.ABC
c6
S(2;l;2;,
A(3;0;-l),
day ABC la
tarn
giac
vuong
can tai A va
SAl(ABC).
Hinli

chieu vuong
goc cua A
len
SB, SC
Ian
lup-t
la
hai diem M,
N va
mat
phang
(AMN)
CO
phuong
trinh
2y
+
3z
+
3
=
0 .
Goi
13 la
diem doi xung voi
A
qu.i
trung
diem
E

cua
BC.
1) Viet phuong
trinh
duong thang SD.
2) Viet phuong
trinh
duong th^ng BC.
J!&igidi.
1) Tu giai thie't
ta
suy
ra
duoc
ABCD
la
hinh
vuong
Ta c6: BD1 (SAB) BD
J
AM, lai
c6
AM
1 SB
nen
suy ra
AM
1
(SBD) hay
ta

suy
ra
dugc
AM
1
AD.
Hoan toan tuong tu,
ta
chung
minh
duoc
AN
1 SD
Suy
ra SD 1
(AMN).
Vay phuong
trinh
SD:
x-2
y
-1
+
2t.
z
- 2
+
3t
B
D

2)
Taco
SA
=:(l;-l;-3) nen phuong
trinh
mat phang (ABC)
la:
x-y-3z-6=0.
Vi
D
= SD
n
(ABC) nen tpa dp cua
D la
nghi^m ciia h^:
Cty
TNHH
MTV DWH Khang
VilT
x =
2
y
=
l
+ 2t
z =
2
+
3t
x-y-3z-6=0

<=>
x =
2
y
=
-l=>D(2;-l;-l)
z = -l
5uy
ra
trung
diem cua
BC
la:
E
2
2
phuong
trinh
BC:
2
^
2
z = -l
Vidu
S.3.,'i.Trong
khong gian Oxyz
,
cho hai duong thang
x-3
y-1 z , x-2

y-l_z-3
di:—-^
= Ig;d2:
—= —= —
La
mat ph3ng (a):
2x + y + z - 7
=
0 .
Duong th^ng A
cat
di
va
da tuong ung
6
A va B,
dong thai khoang
each
tu A
den mat phang
(a)
bang
S.
Viet
phuong
trinh
A
,
biet diem
A c6

hoanh do duong.
Xgigidi.
taco
A 6
di
nen
A(3 + a;l+
2a;-5a),
Bed2
=^
B(2
+
3b;l-b;3
+
2b)
Suyra
AB = (3b
- a -
l;-b
-
2a;2b
+
5a
+
3)
Do
A
each
(a)
mot khoang bang

76
nen
ta c6
<=> <
AB.n„
= 0
d(A,(a))
= V6
AB//(a)
d(A,(a))
= V6
2(3b-a-l)
+
(-b-2a)
+ (2b +
5a
+ 3)
= 0
=
f6
a
= 6
(do
XA
>
0)^^(9.13._30),
X^
=
(-10;-11;31)
b

=
-1
Phuong
trinh
A:
x-9 y-13
_
z +
30
Vi
du
9.3.3.Trong
khong gian Oxyz, cho hai duong thang
x = l-2t
x =
3-k
y
=
2
+
t va
d2 :^
y
=
l
+ 2k.
z =
3-2t
z=2+2k
li

261