Bài tập về tọa độ trong không gian

WWW.ToanCapBa.Net
BÀI TẬP PHƯƠNG PHÁP TỌA ĐỘ TRONG KHÔNG GIAN
1.  !"## #!#
$%&%&%'($)&%&%'(!$%&)&%'*#$%&%&)'"
+,-./0$1'230!#4567
8.#-./0$1'9:;5678.-./0$1'"
9+.8.<=30# 4
·
%
>?%BMD
=
"
2. ./0$1'@?−−?−?A%*30
3. $B'@
> ?
> ? >
x y z+ −
= =
−
",-./;=$C'DE.F=$B'(F6$1'.
59:?*$1'G$C'H.3I=-D96J9:)"
4. 2B !$%&%&?'($)&%&K'(
$>&>&%'(!$L&>&?'".ME.NO.6 *J56
P30!N*"
5. 8.$−>&?&−)'($?&−>&−Q'*
.$1'@R?R−)A%",-.$S'2*T.$1'.
DαU.V@
)
4
Q
α

=
6. 30$B'@
) >
> > ?
x y z− +
= =
−
*8.$?&
−>&>'($%&>@−?'".8.<=30$B'4.6<
DBJUW"
7. 8.$?&−>&>'($%&>@−?'*30
$B'@
) >
> > ?
x y z− +
= =
−
",-30$∆'X=8.O3
0$B'./0$'(:../0$'*Y3
0$B'.Dα4
K
4
Q
α
=
"
8. .6 $>&%&%'($%&?&%'*
$)&%&L'".8.CZ./04C =D./0
$ '"J8J[DC"
9. 30

> ?
> L
$ '@ > ? &$ '@
> ? K
)
x t
x x z
d y t d
z t
=

− −

= − − = =


= −

" 2.$B
>
'*$B
?
'G=",-./0$1'2$B
>
'*$B
?
'"
9" J8J;T9\./0$1'*9./0"
10. (8.<$?&>&%'*30$B'D
@

 >  > 
? > >
− +
= =
−
"
11. ,-.4[O30X=8.<(G*=D
30B"
WWW.ToanCapBa.Net
1
WWW.ToanCapBa.Net
12. (8.<$?&>&%'*30BD
@
 >  > 
? > >
− +
= =
−
",-JGO30X=
8.<(G*=D30B"
13. T(D26]=C" !(9-
C$)&?&L'($>&?&)'(!$)&%&)'"^F*E../;=T-DC" !"_
./0
( )
α
2F*44 "
14. T(*./;=$C'@
? ? ?
? L ? ) %x y z x y z+ + − − + + =
*8.$>&%&%'($>&>&>'",-./

0$1'X=8.(*G./;=$C'H-B*.ID
BJ
)
π
"
15.            T    (    ./  ;=  $C'@
? ? ?
? L Q >> %x y z x y z+ + − + − − =
 *./0$
α
'@ 
? ? >` %x y z+ − + =
",-
./0$
β
'44
( )
α
*G$C'H-B*3ID=
9:
Q
π
"
16. T(8.$>&K&%'($)&)&Q'*3
0
( )
> >
@
? > ?
x y z+ −

∆ = =
−
".T8.<=30$∆'8.6<
DBJUW"
17. a(8.$L&b&−b'($−>%&>)&>'
*./0$1'@RK−`−KA%".8.<Z./0$1'4
<
?
R<
?
T6cUW"
18. a(30
 >  >  L
$B' @
? ) ?
− + −
= =
−
*./;=$C'@
?
R
?
R
?
−>%−?R>%A%",-./0$1'
230$B'*G./;=$C'H.3ID96JUW"
19. (30∆@
  >  ?
> ? >
+ −

= =
−
(
.$1'@?R)−Q−?A%*8.$%&>&)'",-./;=$C'X=
8.(E.=30∆*-d.$1'"
20. (8.$−>&?&−)'($?&−>&−Q'*
.$1'@R?R−)A%",-./0$S'X=(*T$1'
.DαU.V
)
4
Q
α
=
"
21. (8.<$)&?&−?'(./;=$C'@
?
R
?
R

?
−LRL−L−?A%*./0$1'@R?RL−)A%",-
./0$S'X=8.<(=D./0$1'*-d./;=
$C'"
22. (98.$?&%&>'($>&%&%'( $>&>&>'
*./0$1'@R?R?RLA%"
,-./;=X=98.((
*-d./0$1'"
WWW.ToanCapBa.Net
2

WWW.ToanCapBa.Net
23. T(98.$K&e&−>>'($)&K&−L'(
$?&>& −Q'*300$B'@ 
> ? >
? > >
x y z− − −
= =
"f6cT8.<
=$B'4
MA MB MC− −
uuur uuur uuuur
T6cUW"
24. T(D26]=C" !(9-
C$)&?&L'($>&?&)'(!$)&%&)'"^F*E../;=T-DC" !"_
./0$α'2F*44 "
25. T( !"## #!#D
$%&%(%'($>&%&%'(!$%&>&%'* #$%&%&>'"^<(g;Y*=8.O
# #*&1(S*68.;Y=630!* !#41S
44<g"_./0230<g*1S"
26. T(.TME.NO.6
9-$)&%&%'($%&?&%'( $%&%&>'"
27. T(2B !$>&−>&?'($>&)&
%'( $−)&L&>'*!$>&?&>'",-./0$1'X=(4
567 -$1'9:567!-$1'"
28. T(./0$α'@
) ) ? )` %x y z− + + =
*6
8.$L&>&K'($)&%&>'( $−>&?&%'".T8.<=$α'898=24=
T6cUW@
<"< <"< < "<+ +

uuuur uuur uuur uuuur uuuur uuuur
"
29. T
yz
(
( ) ( )
 %&>&? ( >&>&%−
*./0$1'@
%x y z− + =
".T8.<=$1'4.6
<
=ET"
30.  T(<$?&>&?'*30$B'@
? >
> > >
x y z+ −
= =
".Z$B'8.(4.6<]="
31. T(30
( )
> > >
@
> ? ?
x y z− + −
∆ = =
*
./0$1'@
? ? ? %x y z− + + =
",-./;=DE.:.Z3
0

( )
∆
*-d./0@./0*./0$1'"
32. T(30$B'@
?
> ?
x t
y t
z t
= −


=


= − −

*./
0$1'@
> %x y z+ − + =
"^$B
#
'*-=O$B'Z./0$1'".T
8.N=$B
#
'4N68.h$>&>&L'.59:K"
33. T(./0$α'@
? ? %x y+ + =
*68.
$>&%&%'($%&>&%'( $%&)&?'".T8.<(9-:<6]=68.

((
*./0$α'"
34.            T    (      =  <g1S  D
( ) ( )
< K&)& > (1 ?&)& L− −
" .TiS(9-:ig:../0
Q %x y z+ − − =
"
35. T(.T8.S[28.
( )
1 ?& K&`−
X=30X=8.
( ) ( )
> ?
< K&L&Q ( < ?& >`& e− − −
WWW.ToanCapBa.Net
3
WWW.ToanCapBa.Net
36.  T(8. 
( ) ( )
 >& )&% ( >& )&%−
 *
( )
< %&%&a
  
%a >
"ZaW8.g4./0
( ) ( )
g ( <
=D="NV.

a
88J[D" <gT6cUW"
37. T(-.4[O3
0$B'23JO./;=$C'@
? ? ?
? Q >> %x y z x y z+ + + − + − =
9-:$B'
=DD./0$1'@
K ? >` %x y z− + − =
"
38.           T    (    ./  ;=  $C'@
? ? ?
? L L >Q %x y z x y z+ + − − + − =
*30$B'@
> )
> ? ?
x y z− +
= =
" 2.:i
DB=W../0-d./;=$C'*230$B'",-
./0*"
39. 30∆
>
@
> > )
> > >
x y z+ + −
= =
− −
&∆

?
@
> ?
>
x t
y
z t
= − +


=


=

"
j30∆X=8.F$%&)&−>'(G∆
>
T(G∆
?
T"Jk4[
IA
IB

40. T
yz
(30∆
>
@
> ?

? > >
x y z− +
= =
−
&∆
>
@
> > )
> ` >
x y z+ − −
= =
−
"j3=D=O∆
>
*∆
?
G∆
>
T(G∆
?
T"
JBJ∆"
41. ./0$1'@)−eR`RLA%*8.
$>&>&−)'($)&>&−>'".8. =./0$ '4.6
 ]="
42. 
yz
(./0$1'@?R−?RbA%(30
$B'@
> > )

> ` >
x y z+ − −
= =
−
",-30 ∆ =D$1'*U
.V∆G$B'T.8.<6$1'.59:?"
43. Oxyz(./;=$S'@x
?
Ry
?
Rz
?
R?xlQyRLz
lLA%*8.A$%&%&>'(B$>&?&)'",-./0$P'X=A(B
*-d./;=$S'"
44. Oxyz(30
) >
$ ' @
> > ?
x y z
d
− +
= =
−
*
8. A$?&−>&>'( B$%&>&−?'"^ I *8.O$d'./0$OAB'",-
30∆X=I(:.$OAB'*T$d'.Dα9-
K
4
Q

α =
"
45.  ma "## #D6 *.6]=T(#A
 )
?
*
#A# A"J5678.-./0$# # '"
WWW.ToanCapBa.Net
4
WWW.ToanCapBa.Net
46.  2 !"## #!#DA!A?*
·
%
! Q%=
"^<*
=8.O#!#"J567-./0$!<'(9-: #
=D./0$!<'"
47.  DC" !D6 !*PT
 )(  ?= =
"
N-==DO8.CZ./0$ !'nE.O
.6 !(DP./0$C '*$ !'9:Q%
%
"^<*=
8.C *o*8.P30<./0$C!'"J8J
[2B o"
48.  DC" DT9Z C =D./0 $ '(
· ·
  = = α
(!A(

·
C! =β
!*=8. "</0X=*=
DC!GC(C ;YT<(g"J8J[DC"<gH(p*
β"
49. Oxyz(30∆@
) ? ?
> > >
x y z+ − +
= =
−
(./
0$P'@xR?y lzRKA%*./;=$S'@x
?
Ry
?
Rz
?
R?xlQyRLzA
%",-./0$Q'=D$P'(44∆*-d
./;=$S'"
50. T Oxyz(D26]=S"ABCD
S$>&?&−?'(B$)&?&−>'(D$>&%&−)'"^I*E../;=T-DS"ABCD"
_./0
( )
α
2BI*44AC"
51. Oxyz(.6ABCDA$>&)&?'(B$q>&?&)'*
C$q?&%&>'",-30∆X=ME.O.6ABC*
=D./0$ABC'"

52. Oxyz(30∆@
> ? >
> > ?
− − −
= =
−
x y z
*
8.A$%&>@−?'(B$?&−>&>'".8.C=30∆4.
6ABCDBJUW"
53.  D26]=C" !D6*=T(T9ZCA
?"^<(g;Y*=8.OC!* (1*8.[2<X=
=8.OC"J4ODP30C*g1"
54.
 DC" !D6 !*T(
·
%
Q%BAD =
" T9Z
C=D./0$ !'*CA"^<*=8.OC "J
4ODP30<*C"
55. Oxyz(9[8.A$Q&−Q&Q'(B$L&L&L'(C$−?&>%&
−?'*S$−?&?&Q'" 2.O(A(B(C*9[iO.*
-==DO S Z./0$OABC'nE. I O OABC"J
56P30SO*AC"
56. Oxyz(98.A(B(C;YBZ6
Ox(Oy*Oz4./0$ABC'X=O*=X=8.M$>&?&
)'"f6c68.A(B(C88J[2BOABCT6cU
W"
WWW.ToanCapBa.Net

5
WWW.ToanCapBa.Net
57. Oxyz(./0$P'@xRz−)A%($Q'@yRzR
KA%*8.
$>& >& >'A − −
".68.MZ$P'(NZ$Q'4MN
=D=-O$P'($Q'*A*=8."
58. Oxyz(98.
$ >&%&>'( $?& >&%'( $?&L&?'A B C− −
*./
0 
$ '@ ? ? %x y z
α
+ + + =
"  .      8. M Z  $α'  4    98=  2
? ? ?
T MA MB MC= + +
T6cUW"
59. Oxyz(8.A$%&%&−)'&B$?&%&−>'*./0
$P'@)x−y−zR>A%".8.C:.Z$P'4ABC.6]="
60. Oxyz(.6ABCDA$)&>&%'(B:.Z./
0Oxy*C:.ZaOz".68.B(C4H$?&>&>'*M
E.O.6ABC"
61.  Oxyz(-30∆ X=8.
A$%&>&−?'(=D30
) ?
$ ' @
> > >
x y z
d

+ -
= =
-
*T./0$P'@
?xRy−zRKA%.D)%
%
"
62. Oxyz(98.M$%&−>&−?'(N$−>&>&)'* I$%&%&
?'",-./0$P'X=M(N*U.V567I-
$P'T6cW"
63. A$>&q>&%'*30
d:
1 1 1 1 1
' :
2 1 2 1 2 1
x y z x y z
va d
+ - + + -
= = = =
-
• <r@d*d’s=
• ,-30∆X=A*Gd*d#
64. (30$B'@
5 7
2 2
x y
z
+ -
= =
-

*
8.M$L&>&Q'"j30$B'G./;=$C'E.*<T8.A, B4
ABAQ",-O./;=$S'"
65. 30@
1
1
: 2
2
x t
d y t
z t
ì
ï
= -
ï
ï
ï
ï
=
í
ï
ï
ï
= - +
ï
ï
î
*
2
: 1 3

1
x t
d y t
z t
ì
ï
=
ï
ï
ï
ï
= +
í
ï
ï
ï
= -
ï
ï
î
"_30$B'X=<$>&%&>'*G53
0B
>
*B
?
"
66. 30@

1
1

: 2
2
x t
d y t
z t
ì
ï
= -
ï
ï
ï
ï
=
í
ï
ï
ï
= - +
ï
ï
î
*
2
: 1 3
1
x t
d y t
z t
ì
ï

=
ï
ï
ï
ï
= +
í
ï
ï
ï
= -
ï
ï
î
"
_./;=D3J*T=D=OB
>
*B
?
"
WWW.ToanCapBa.Net
6
WWW.ToanCapBa.Net
67. aT./0$1'@?lRR>A%
*30
1 2
1 2 3 1 1 2
( ) : ;( );
2 1 3 2 3 2
x y z x y z

d d
- + - + - -
= = = =
",-
30$∆'4
4$1'&=D$B
>
'*G$B
?
'ToD*9:)"
68. aT30
1 2
3 1 2 2
( ) : ;( );
1 1 2 1 2 1
x y z x y z
d d
- + - +
= = = =
- -
"<30$B'X=8.$>&?&)'(
G
 30$B
>
'T*G$B
?
'T " 2.:*=8.O
T
69. aT./10$1'@RRR)A%(
30$B'@

1 2
1 2 1
x y z+ -
= =
-
*6 8.$)&>&>'($`&)&b'( $?&?&?',-
./0$S'2$B'*44$1'
70. aT./10$1'@)R?qRLA
%*8.$L&%&%'*$%&L&%'"^F*=8.O".T
8.O30./0$1'*6cT8.h4
hF⊥$1'(t3h6]=[T*./0$1'
71. E( đường thẳng (d) :
3 4 3
1 2 1
x y z- - +
= =
-
và mặt
phẳng (P): 2x + y + z = 0 . Gi A là giao điểm của (d) và mp(P) ,viết phương trình
của đường thẳng
$ '∆
đi qua điểm A, vuông góc với đường thẳng (d) và nằm trong
mặt phẳng (P).
72. a(Cho đường thẳng d:
1 2
1 2 1
x y z+ -
= =
-
và

mặt phẳng (P): 2x – y – 2z – 2 = 0
Viết phương trình mặt cầu (S) có tâm I nằm trên d cách (P) một đoạn bằng 2 và
mặt cầu (S) cắt (P) theo một đường tròn giao tuyến có bán kính b:ng 2.
73. a(630
$B'@
1 1 1
1 2 2
x y z- - -
= =
*$B#'@
1 3
1 2 2
x y z+ -
= =
-
2U:$B'*$B#'G=",-./0$1'2$B'*$B#'
74. a(30B
>
@
1 1 2
2 2 1
x y z- - -
= =
(
30B
?
*=-O./0$1'@KlQlQR>)A%*$S'@q
QRQl`A%
2U:$B
>

'*$B
?
's="
^ *8.$B
>
'*$B
?
'".T68.(;Y=$B
>
'($B
?
'4
.6 ET *DBJ9:
41
42
WWW.ToanCapBa.Net
7
WWW.ToanCapBa.Net
75. =T(68.$q>&q>&%'($>&q>&?'(
$?&q?&>'(!$q>&>&>'"
JD*56P30* !
^54u.$1'X=!*G9a((2T68.<(g(16
[4!*ME.O.6<g1"NV-O.$1'"
76. aT30s=

1 2
2 3 4 1 4 4
( ) : ; ( ) :
2 3 5 3 2 1
x y z x y z

d d
- - + + - -
= = = =
- - -
.T8.O3=D=BO$B
>
'*$B
?
'*
3D=BD"
77. (30$!'@
2
2
2 2
x t
y t
z t
ì
ï
= - +
ï
ï
ï
ï
= -
í
ï
ï
ï
= +

ï
ï
î
^
D
*30X=8.$L&%&q>'44$!'*F$q?&%&?'*
-==DOZ$!'"6./0X=
D
(V-
O./0$1'D56-$!'*W"
78. (-30$B'X=
8.$)&q?&qL'(44./0$1'@)l?l)l`A%*G3
0
D
@
2 3
4 2
1 2
x t
y t
z t
ì
ï
= +
ï
ï
ï
ï
= - -
í

ï
ï
ï
= +
ï
ï
î
"
79. Oxyz(V6cTE.*96J
3IT-.
6ABC(9-A$q>&%&>'(B$>&?&q>'(C$q>&?&)'"
80. Oxyz( 98.A$%&>&?'(B$?&q?&>'(C$q?&%&>'"
,-./0$ABC'*.8.M=./0?xR?yR z l)A
%4MAAMBAMC"
81. (8.$K&L&)&'&*630

( ) :
2 3 1
m
x y z m
d
-
= =
*
1
( ) :
2 3 1
x y z
d
-

= =
-
"
.8.∈$d)*4[M.868.=$B
.
'=6]=&
82. (630

1 2 3
( ) :
2 3 1
x y z
d
- - -
= =
*
3
( ') :
2 3 1
x y z
d
+
= =
-
"
,-./;=E.F∈$d’)(96J9:
3 3
*-d(d)
83. (630∆
>

(∆
?
*.$1'D@∆
>
@
> > ?
? ) >
x y z+ − −
= =
(
∆
?
@
? ?
> K ?
x y z− +
= =
−
(.$1'@?−−KR>A%
>+ .∆
>
*∆
?
s="J56P?30W"
WWW.ToanCapBa.Net
8
WWW.ToanCapBa.Net
?+,-30∆=D.$1'(t3G5∆
>
*∆

?
"
84. (630B
>
@
? > %
> %
x y
x y z
+ + =


− + − =

*B
?
@
) ) %
? > %
x y z
x y
+ − + =


− + =

>+ .B
>
*B
?

t0*-.$1'2B
>
*B
?
"
?+.8J;T9\.$1'*9./0"
85. (630B
>
@
? ? %
) %
x z
y
+ − =


− =

*B
?
@
?
>
?
x t
y t
z t
= +



= −


=

>+ .B
>
*B
?
G==D="
?+,-3=D=OB
>
*B
?
"
86. (630B
>
@
e ?) %
L >% %
x z
y z
− + =


− + =

*B
?
@

? ) %
? ? %
x z
y z
− − =


+ + =

>+,-.$p'2B
>
*44B
?
"J56PB
>
*B
?
"
?+,-30∆44a*G5B
>
*B
?
"
87. (L8.$%&−>&>'($%&−?&%'( $?&>&>'(!$>&?&>'
>+,-.$p'2*=D.$ !'
?+.8.<=30!*8.g=302a
4<g*=D=O30*"
88. (30B@
> > ?
? > )

x y z+ − −
= =
*.$1'@−−−>A%
>+_JGO30∆X=$>&>&−?'44$1'*=
DB"
?+_./;=$C'DE.=B(96J9:)
)
*-d$1'"
89. (30B@
K ) >
> ? )
x y z− + −
= =
−
*.$p'@?R−−?A%
>+.8.<OB*$p'",-30∆:..$p'
X=<*=DB"
?+ 8.$%&>&>'"NV.8.4.$p'*./=MO
T0"
90. (ma2"###$?&%&%'($%&L&%'(#$%&
%&L'
>+.68.#(#",-./;=$C'X=L8.(((#"
?+^<*=8.O"<$1'X=<=D#*G(#
;YTg(h"JB*Thg"
91. jZ=D
 !"## #!##$%&%&%'(#$%&?&%'(!#$?&%&%'"^<(g(1(SH2M*
=8.O6T!# #( ##(#(!"
>+.-=O Zg"
?+ <r30<S*g1n:.../0*JB
J26<g1S"

N!@^⇒ #$?&?&%'($%&%&?'($%&?&?'(!$?&%&?'( $?&?&?'
92. (9[8.$L&L&L'&$Q&qQ&Q'& $q?&>%&q?'*C$q?&?&Q'"
WWW.ToanCapBa.Net
9
WWW.ToanCapBa.Net
2. *>*2.CF=D./0
$ '$F*E.O'
J8JODC" *56P?30C*

^<*=8.C(./0$<'GC Tg(JBJ26
<g"
93. aTj]6(30
>
@
> ? >
x y z
= =

(
?
@
> > >
> > )
x y z +
= =


2.30
>
*

?
s="

,-./0$1'230
?
*T30
>

.D)%
%
94. 8.$)&?&)'*30
>
? ) )
@
> > ?
x y z
d

= =

*
?
> L )
@
> ? >
x y z
d

= =

" 2.30d
1
; d
2
*8.n:.
../0"f6cT6i* O.6 9-B
>
2
3N*B
?
23==- <O.6 "
95. 30B@
> ?
> > >
x y z +
= =
*./0$1'@?
Rl?R?A%"_./;=$C'DE.:.ZB(-d./
0$1'*X=8.$?&q>&%'
96. Trong không gian với hệ tọa độ Oxyz cho điểm A(10; 2; -1) và đờng thẳng d có
phơng trình
)
>
>?
>
==
zyx
. Lập phơng trình mặt phẳng (P) đi qua A, song song với d
và khoảng cách từ d tới (P) là lớn nhất.
97. Trong không gian với hệ tọa độ Oxyz cho điểm A(10; 2; -1) và đờng thẳng d có

phơng trình





+=
=
+=
tz
ty
tx
)>
?>
. Lập phơng trình mặt phẳng (P) đi qua A, song song với d và
khoảng cách từ d tới (P) là lớn nhất.
98. jZ6=D30@
$B'

> ?
L K
=


= +


= +

*$B#'

> ?
)
=


=


=

<r30$B'*$B#'G="
,-JGO/30E6ODT9\$B'*
$B#'"
99. jZ6=D.$1'@l?Rl?
A%*30@$B'
> ) ?
> > ?
+ +
= =

*$B#'
> ?
?
>
= +


= +

= +

WWW.ToanCapBa.Net
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WWW.ToanCapBa.Net
,-.4[O30$
∆
':../0$1'*G5
30$B'*$B#'" <r$B'*$B#'s=*J56P
d"
100. aT8.$>&%&>'($?&>&?'*
./0$S'@
R?R)R)A%"_./0$1'X=(*=D
$S'"
101.  !"## #!#

DT9:?"^<*=8.O
T!(g*
E.= #!#!"J96J./;=X=68.( #(<(g"
102. Oxyz(8.A$q)&K&qK'(B$K&q)&`'*./
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103. Oxyz(./0$P'*30$d';
YD@
$1'@?x−y−?z−?A%& $d'@

> ?
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x y z+ −
= =
−
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.DUW"
104. (./;=$C'*./0$1'D
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105. 98.$>&K&L'($%&>&>'( $>&?&>'"
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106. ,-./0
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107. 8.
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*./0
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108. T8.$>&%&%'&$%&?&%'& $>&)&>'" <r@
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WWW.ToanCapBa.Net
11
WWW.ToanCapBa.Net
109. T8.$>&%&%'&$%&?&%'& $>&)&>'" <r@
(( 0**.TE.3IT-x "
110. 6H
a
r
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b
r
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111. .6 D$>&?&−>'($?&−>
&)'( $−L&`&K'"JB*3E6y7i"
112. (./;=
$C'@
?
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∆
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x y z−
= =
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*∆
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s="
• ,--BO./;=$C'(9--BD44
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113. (./;=
$C'@
?
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∆
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x z
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s="
• ,--BO./;=$C'(9--BD44
30∆
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114. zP !"## #!#
D=[($&%&%'(!$%&&%'(#$%&%&9'(9{%"^<
*=8.T #"J8J[2B!#<H*9*6ci
4[
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b
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115. 8.$−>&Q&Q'($)&−Q&−?'"
.8.<=.$'4|<R<T6cUW"
116. )8.$>&%&−>'($>&?&>'( $%&
?&%'"^^*E..6 "
• ,-30^"
• ,-./;=$C'X=L8.((( "
• ,-6.=D30^*-d./;=
$C'"
117. )8.$>&%&−>'($>&?&>'( $%&
?&%'"^^*E..6 "
• ,-30^"
• ,-./;=$C'X=L8.((( "
• ,-6.=D30^*-d./;=
$C'"
118. ?.@
WWW.ToanCapBa.Net
12
WWW.ToanCapBa.Net
$1'@Kl?RKl>A%*$S'@lLleR>?A%
_.$α'X=[(=D.$1'*Y.
$S'.DLK

"
119. (8.$%&>&?'*30@
B
>
@
1 1

2 1 1
x y z− +
= =
−
B
?
@
1
1 2
2
x t
y t
z t

= +

= − −


= +

,-./0$1'X=(t344B
>
*B
?
".
68.<ZB
>
(gZB
?

4)8.(<(g0*"
120. ( !"## #!#
9-$%&%&%'($>&%&%'(!$%&>&%'(#$%&%&>'"_.$α'2
30 !#*T.$#!#!'.DUW"
121. _.$α'X=8.$?&−>&%'($K&>&>'*56
78.<
1
0;0;
2
 
 ÷
 
-.$α'9:
7
6 3
"
122. L8.$)&>&?'($−>&−)&%'(
$L&%&−)'*!$?&?&−>'"
J5678.-.$ !'"
].ON*-==DOZ.$ !'"
,-.$1'X=*=D30 !"
.8.h*ME.O.6 !"
123. )8.$&%&%'($%&9&%'( $%&%&'(9(*94[B|
*=U.V
?
R9
?
R
?
A)"f6c(9(45678.

$%&%&%'-.$ '*W"
124. ?8.$>&?&)'($−>&L&?'*.@
$1'@?lQRLR)A%
$S'@lRR>A%
.8.hO30.$1'".8. :.Z
.$S'4.6 *.6]="
125. ,-./0$α'X=8.<$b&>&>'*G6((
;YT(( 4RR D6cUW"
126. ,-./0X=8.<$b&>&>'G6((T(
( 48J2B D6cUW"
127. Oxyz(./0$P'*30$d';
YD@
$1'@?x−y−?z−?A%& $d'@
> ?
> ? >
x y z+ −
= =
−
>",-./;=DE.=30$d'(6./0$P'.
59:?*G./0$P'H=-*3ID96J9:)"
?",-./0$Q'230$d'*T./0$P'.
DUW"
WWW.ToanCapBa.Net
13
WWW.ToanCapBa.Net
128. %30@
!
>
@
2 1

1 1 2
x y z

= =

(!
?
@
2 2
3
x t
y
z t
=


=


=

,-./;=D3J*T=D=O!
>
*!
?

129. %30B
2 1
4 6 8
x y z +

= =

*
8.$>&q>&?'($)&qL&q?'".8.FZ30B4FRFT
6cUW
130. T./;=
? ? ?
$ ' @ ? Q L ? %S x y z x y z+ + + =
"
,-./0$1'446Os
$>&Q&?'v
r
(=D
./
0
$ '@ L >> %x y z

+ + =
*-d$C'"
131. T./;=
? ? ?
$ ' @ ? Q L ? %S x y z x y z+ + + =
"
,-./0$1'446Os
$>&Q&?'v
r
(=D
./
0
$ '@ L >> %x y z

+ + =
*-d$C'"
132. aT8.$>&%&>'($?&>&?'*
./0$S'@
" R?R)R)A%"_./0$1'X=(*=
D$S'"
133. !"## #!#

DT9:?"^<*=8.O
T!(g*
" E.= #!#!"J96J./;=X=68.( #(<(
g"
134. Trong không gian với hệ tọa độ Oxyz cho điểm A(10; 2; -1) và đờng thẳng d có
phơng trình
)
>
>?
>
==
zyx
. Lập phơng trình mặt phẳng (P) đi qua A, song song
với d và khoảng cách từ d tới (P) là lớn nhất
135. Trong không gian với hệ tọa độ Oxyz cho điểm A(10; 2; -1) và đờng thẳng d có
phơng trình

+=
=
+=
tz
ty
tx
)>
?>
. Lập phơng trình mặt phẳng (P) đi qua A, song song với d và
khoảng cách từ d tới (P) là lớn nhất.
136. a(30@
> )
> > L
x y z
= =
*8.<$%&q?&%'",-./0$1'X=8.<4
430t356P30*./0
$1'9:L"
WWW.ToanCapBa.Net
14
WWW.ToanCapBa.Net
137. (30B
>
@
> > >
? > >
x y z+ − −
= =
−
&B

?
@
> ? >
> > ?
x y z− − +
= =
*./0$1'@qq?R)A%",-
JGO30∆(9-∆:.Z./0$1'*∆G
30B
>
(B
?
"
138. T(8.F$>&K&%'*30

>
@ L
> ?
x t
y t
z t
=


∆ = −


= − +

&

?
?
@
> ) )
x y z−
∆ = =
− −
,-.4[O30BX=8.F*G530
>
∆
*
?
∆
,-./0$
α
'X=8.F(44
>
∆
*
?
∆
139. T=D(30@

? > % ) ) %
$ ' &$ }'
> % ? > %
x y x y z
x y z x y
+ + = + − + =
 

∆ ∆
 
− + − = − + =
 
" 2.:30$
∆
'*
$
}∆
'G=",-JGO/30E6O6
DT9\$
∆
'*$
}∆
'"
140. T(8.$>&?&q>'($`&q?&)'*
30BD
? )
? $ r'
L ?
x t
y t
z t
= +


= − ∈


= +


".ZBP8.<4
|567<-**UW"
141. aTOxyz L8.A$>&q>&?'(B$>&)&?'(
C$L&)&?'(D$L&q>&?'*./0$P'D@
%? =−++ zyx
"^A#*
ZdOAZ./0Oxy"^$S'*./;=X=L8.A#(B, C,
D"f6cTE.*96JO3I$C'*O$P'*$S'"
142. aTOxyz
( )
%K?@ =+−+ zyxP
*30
)>
?
)
@'$ −=+=
+
zy
x
d
(8.A$q?&)&L'"^
∆
*30:.Z$P'X=
8.O$d'*$P't3=Dd".Z
∆
8.M4
56AMGW"
143. (9./0@
$1'@?lRR>A%($S'@lR?R)A%($r'@R?l)R>A%

*30
>
∆
@
?
?
−
−x
A
>
>+y
A
)
z
"^
?
∆
*=-O$1'*$S'"
,-30$B'=D$r'*G530
>
∆
(
?
∆
"
144. (8.<$>&q>&>'
*30
>
$ '@
> ? )

x y z
d
+
= =
− −
*
> L
$ }'@
> ? K
x y z
d
− −
= =
WWW.ToanCapBa.Net
15
WWW.ToanCapBa.Net
2.@8.<($B'($B#'n:.Z../0",-./
0D"
145. h" 30





=
=
=
∆
>
?@

z
ty
tx
*8.
'>(%(>$ −A
.68.o*~=30
∆
8.6o~*.6]="
146. " ./;=$C'@
( ) ( )
b?>
?
?
?
=+++− zyx
"
_./0$1'=D30@
??
>
> −
=
−
=
zyx
*G
./;=$C'H3ID96J9:?"
147. 30$B'@
 
 >
 

=


= −


= −

*?.$1'@R?R?R)A%*$S'@R?R?
R`A%
" ,--=O$B'Z$1'
9" _"./;=DE.F=30$B'*-d./
0$1'*$S'
148. (630@
( )
>
 >
B @  L ?
 ) 
=


= − +


= +

 *
( )
?

 )=
B @  ) ?=
 ?
= −


= +


= −

2.:$d
>
'*$d
?
's="
,-./;=$C'D3J*T=D=O$d
>
'*
$d
?
'"
149. ( 98.$>&K&L'($%&>&>'( $>&?&>'"
.8.!=304B*T0 !U
W"
150. ( 30D@
151.
>
? )
@ >

) ?
x z
d y
− +
= + =
?
)
@ ` ?
>
x t
d y t
z t
= +


= −


= −

,-30GB
>
*B
?
t3X=8.<$)&>%&>'"
152. jZ6=D30@
$B'
 
 > ?
 L K

=


= +


= +

*$B#'
 
 > ?
 )
=


= − −


= −

" <r30$B'*$B#'G="
9" ,-JGO/30E6ODT9\
$B'*$B#'"
153. jZ6=D.$1'@
WWW.ToanCapBa.Net
16
WWW.ToanCapBa.Net
l?Rl?A%*30@$B'
 > )   ?
> > ?

+ − +
= =
−
*$B#'
 > ?
 ? 
 > 
= +


= +


= +

,-.4[O30$
∆
':../0$1'*G5
30$B'*$B#'" <r$B'*$B#'s=*J56P
d"
154. T(./0$1'@?llKR>A%*
30
B
>
@
> > ?
? ) >
x y z
+ − −
= =

(B
?
@
? ?
> K ?
x y z
− +
= =
−
,-30B=D$1't3G30B
>
*B
?
"
155. T30B@
) ? >
? > >
x y z
− + +
= =
−
*./0$1'@
RRR?A%"^<*8.OB*$1'",-30
∆
:../0$1'(=DBt35.V567
<
∆
9:
L?
"

156. ma2ABC.A’B’C’DW56T]=9:a. ^ M *=
8.OAA’.J8JO[2BBMB’C’Ha *2.:
BM =DB’C.
157. jZ6=D ./0
( )
@ ? ? > %P x y z− + − =
 *630 
>
> )
@ (
? ) ?
x y z
d
− −
= =
−

?
K K
@ "
Q L K
x y z
d
− +
= =
−
.8.M=d
1
(N =d
?

4MN44$P'*30
MN6$P'.59:?"
158. Oxyz, V6cTE.*96J3
IT-.6ABC(9-A$q>&%&>'(B$>&?&q>'(C$q>&?&)'"
159. (30@
>
$ '@
> > ?
x y z
d = =
*
?
> >
$ '@
? > >
x y z
d
+ −
= =
−
"
.68.M=
>
$ 'd
*g=
?
$ 'd
430MN4
4./0
( )

@   l     ?%>% %P x y z+ + =
B*TMN9:
?
"
160. Oxyz( 98.A$%&>&?'(B$?&q?&>'(C$q?&%&>'"
,-./0$ABC'*.8.M=./0?xR?yR z l
)A%4MAAMBAMC"
161. Oxyz(V6cTE.*96J3
IT-.6ABC(9-A$q>&%&>'(B$>&?&q>'(C$q>&?&)'"
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162. 30B
>
@
> > ?
x y z
= =
&B
?

> ?
>
x t
y t
z t
=


=

= +

*
8.<$>&?&)'"
>",-./02<*B
>
&.<
#
[2<X=B
?
"
?".
> ?
&A d B d

4GW"
163. 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 :
>

K
)
?
?

+
==
z
y
x
.
Viết phơng trình mặt phẳng
'$

đi qua d và tạo với d một góc
%
)%
164. 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 :
>
K

)
?
?

+
==
z
y
x
.
Chứng minh rằng hai đờng thẳng đó vuông góc với nhau. Viết phơng trình mặt phẳng
'$

đi qua d và vuông góc với d
165. (30@

?
K
>
>
)
L
@
>

+
=


=

zyx
d

>)
)
>
?
@
?
zyx
d
=
+
=


,-./;=D96JUW-d530B
>

*B
?
166. (68.$>&%&%'&$%&?&%'& $%&%&q?'.8.#[
2
X=$ '"
167. ,-3=D=O304=@
> ?
> ?
> ?
B @ &B @ >

? > >
)
= +

+

= = = +



=

168. (8.<$?&>&%'*30B@
> >
? > >
+
= =

",-JGO30X=8.<(G*
=D30Bvà tìm toạ độ của điểm M đối xứng với M qua d
169. Trong không gian với hệ toạ độ Oxyz cho hai điểm A(1;4;2),B(-1;2;4) và đờng
thẳng

:
> ?
> > ?
x y z +
= =

.Tìm toạ độ điểm M trên

sao@
? ?
?eMA MB
+ =
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