Bài tập hình giải tích trong không gian 20162017

I'

'tiww &,oeL ?"*tt"5

-

0n rAp Hir.Ju Hoc clAt rics TRoNG xs0xc GIAN
t.
I

vtrr

PHLPgNG

D4ng 1r

Ccr

rniNn ruAr PHit'tc

bin

B[-1;1;3J va mat oh'Ang [P):
83. Trong kh6ng gian vsl h6 toa do Orvz , cho hal di€jm AQ;a \'vu6ng
g6c vcri m5t phang [P] D/s:
vi
- -?,,-)- -5:0.Vl6t phuong trinh mEt phing [Q) di qua hai di6m A, B
,Qt,2A-3;-11:0.
phuong trinh mit phang [P] di qua har didm '{r 2: 1: 3 ' B(1: -2:1
HT 84. Trong kh6ng gian voi h6 tga oo Oryz , vidt

HT

\

J,

t

a4

T

vi

song song voi duong

thing a

v

:

- _i _
-')r
D

z

Dls:(Pl. ii.c- j1, .'r- -9:Lr'
HT 85. Trong kh6ng gian v6i

:-)
i,,-l
-, : ,

, .r'l
t-''
:t:
HT

h0

.d''

+

D+

tQa d6 Oryz,
:! -,\ :=
'-1
693

cho

2

du&ng

thing (4) vir (d.1c6 phuong trinh:

Lip phuong trinh mit phing IPJ chria [d,)

vi

(dr)

'

D/s:{P): Y+Y-52+10=0
mat ciu tS) c6 Phuong trinh:
Orqz , cho
dd
86. Trong kh6ng gian voi hC tqa
vec rry : : , tl::
,' - !t't + z2 *2t+6y 12- 2:A Viert phrrong tr)nh m5t nhang (P) song song vor gia cua
ru6nggocvmmitphang 'r i - l, -: - II -- L vAtiepxircvdi (S
Dls:(P):2.r -y -r2z:3:l: hoac (P): 2r - l-22 -21: tl'
'

HT

g7.

Trong khong gian v6i

I :+
,,.,
-1r:J
phang

:1.j.

hO

tca de 0:;:

cho drem lvi[1;

-i;

1.,

r: -al diron3 thing ir],; ;

'=

t -t - -,

p\u:ng r-r-lir
Chfrngminh ringdidm i1. d.,.,?, cirng ndm tr0n md?m5.npn5ng. 1r;!"

d6. D/s: t: r2Y -.2 r

?

:

0

'-

lr-rit-

'

Dang 2: Phucrng trinh mit phi'ng li0n quan t6i mit chu
HT

88.

Trong khdng gian vcYi h0 toa

.:) .- !-l) - .;l - ?x - 2,' *'\z * 2 : 0.
xirc vdi mit clu [S1.
(P), :t..22- 3-2Vi:t;
Dls:

dO

Oru:

LAp phr-rong

,

cho dr-rong t'hing

__:r, "!_3
va nrat ciLi

d: t;:
T - -

trinh mat ph;ng (PJ song song v&i d vi

hodc

(P),

trLLC

[S1:

0x, dbng thcn tieP

y-22-3- 2r,5:0'

vAmitphing[P]
Trongkhdnggianvo,rh0toa dO Oru:,chomitciu[SJ: rr -!2-t2-2''-1';-]:0
mitphing [PJ vatrep x'ic
..+ j -3:0.vidtphuongtrinh mitphing [Q) dr quadidrn '1,Ii3:i: -1) vuonggocvoi
vryi mat ciu [Si.
:
D/s: (Q): 2t + 9 -2:-9 : O Hoqc (Q): l:t *7a --lz-9 0

HT89.

Cdu hoi

yri

tvqno

tLr:

(Sf , 12 = y2 + :2' -2r

-

L'J

DS:

- iz i5 : 0' 1P' : 2r - I -6'-5 :
(Q'; 2r'r29 - z-6:a

hoqc (O)

C''11(l:1;2)'

I11r-iOy..22-5:a'

Trongkh6nggianv6i hCtoa dd Orlqz,chomitcau[S]: 12 =92 -'2-2'*1'l*22
c6 bin kinh r :
irinh mit phdng toi.no, truc ox r,) cit mit c;u [S] theo m6t dudng trdn
Dl* Q):Y - 2z = 0.

HT90.

HT91.

Trongkhonggianv6ih6toa dd oiy:,cb.omitclu[S]: 12 -!'2

6>v

arrrrrrflrr

an\r

cE I.t

-3:0
3

Vi6tphugng

'

-:2 +2r-2y-22*7:0

viduongthing

.EFN

Pase 27


-;lr 'ffiTil.fr
ur

'

\

BJ s*.

-

l{*

HT113.
vA

it

r - 6: u

\1*

:t-e

Trongkh6nggianroa do O:tu::. chocacdi€:n l(ll.

/

.r *211

lP;:

0.

TRiNH DI.IONG THANG

:

thing
,.

d

.

.:',)

L,_1

j

?

_3

115. Ti'ong kh6ng gian viri h6 toa d6 Oxyz, cho dudng thing [d) c6 phlrnng lr:hl: { - - i : tr : -1 -'2t
: - l -:- t [t = ,q] vi m5t p]ring [PJ: 2.r - u -2: -:] - il Vi6t phLrong trinh tham sd cua d,;ong thing A narn trOn
,

vi

[P], cat

D/s:

vu6ng g6c v&i (dJ.

l.i

J.

:

I

j,r7
I

l::

rIT

PHLTCYNG

trinh clucrng thing bing c6ch xic tlinh vects chi phuong

,.-')
t ,,
mai phang P
114. Trong kh6ng gian vdi h€ toa dO Oxyz cho cluong thang ,l ' ; : -11.1
,i .l) : - i : t) . Vrdt phuong trinh duong thang -\ di qua -111: 1: -2 , song song voi mat phing {P; v) vudng gdc

vdr drrlng

HT

j

ai

VIET
HT

!-.

ir;.Bir:2: i) .Vi6tphuongtrinhmltpheng[P]quaA,B

ll:

cho tam gi;ic ABC c6 di6n lich bans

- )z- 3 :

Dang 1: Vi6t phuong

l":

*--!,.

).--

cit truc 0z tar M/sao

DS:

tS) &fu" ,t *taivz. w

116.

-;J

t

-

'

.

Tror-rg kh6ng gian vcri h6 toa d,6 Oxltz, cho di0m

phuirng tr)nh cfra dud'ng thlng d di qua tli6m

1"1,

cit vi

M[2; 1; 0] r,i ducrng thang

Lip

vuOng goc vai a

Ox1,2,61'r" mitphing iPl: x + 2y-Zz + I = 0 vahai diem.ii(1;7; -lJ.
(Dl
phuong
tliang
Ia hinh chidu vu6ng goc cua duong thing AB trOn [P).
trinh ducrng
8(4;2;0). LAp
. Gqi (U la mdt phdng qua A, B va vulng goc v1i (P)
= (U, Bx + 7x + i1'z - 46 = 0

HT117. Trongkh6nggianvoihEtructoadij

iD)
HT

',
= (P)

118.

Q) suy ra

phLrcrng trinh {D)

Trong kh6ng gian vdi hC toa d6 Oxyz, vidt phrrong trinh hinh chi6u vu6ng g6c cua dLrong thing

1..

n-

t\

" ..
l,'{ .'''-.
] -'i-t
t''
["

:

|t.t.:++tllt1

t.

Ef

s:1:

trdnmltphing P.r.ail

': r-5-0.

r1r,

lr:!-t,lr.
,,
l"
l. : i- tot
t

Cdu hoi tvong tLr

a) V1i

d

.r

-1_ r-1

[':

r

:
1!t

2 -- 29rrt

I

t^
I"

HT

-.)'1,r,

) - Jltt,

119. Trong kh6ng gran voi he tga dO O.x,-tz, ggi A, B, C lin }trot giao didm c0a mat phing
P : 6r -2,t= il: -6:0 r,6i 0x 01t,Oz. LApphuungtrinhdulngthangddi quatAmduongtronngoaltieptam
grAc ABC

dbng thoi vu6ng g6c vdi

mit phing (Pl.

Hoc v0 Bg - cHUyEN cAtt sE oEN sEI{

Page 31


Ir:i -:

Cl

l:t
.J : :r
D/s: d: ].,,
t') - Zt
I

L

/

-r

I

- I -

.

rlr

120. Trong kh6ng gian vdi h6 toa d6 ?xyz,
'-)
, r-l
u-l
-',_
2
2

IIT

phing

vdi dudng thing

IABCJ vir vuong g6c

' ')
L2

;

-l
2

cho

dudng thing

d.

-1

-I1

trinh dudng thing Ii6n quan il6n m6t

Dang 2: Vi6t phuong

3 didm A(L:2:-\).8(2:l:1;:C(():1:2i v)

thing khic

Clucrng

HT121. Trong kh6ng gian v6i hC toa d6 Oxyz, cho didm M[2; 1; 0J vi du]ng thing d c6 phuong trinh
1
,i , "' - : {+ : :- Vi6t phuong trinh cua duong ihing 5 di qua drdm IvI, cit va vu6ng g6c vdi duong thdng d
lr
1

vi

tirrr toa d6 drdm M' d6i xirng voi M qua d.

:t |
--'
-l
-l

'-j=
D/s:J.
'
t

Cdu hiti

tvongt?':

a) iv[i-4.-2r-1

H'l'1,22.

l/'lll,l -: -1,
;i ;j'
-;i

):11

,- ,,

)

-. -

- ')

clLl'rrng

')

thlng d l---: - !-)

i-r'ong xhong gian v6'i h6 toa cl6 Oxyz, cho dLrong

2z r 2 - 0. l-Ap phiro'ng tnnh

z-i)

.\

DS:

_1

1

- :

;t22

rhang,l song song vdi nr5t phing [P) dr qua

) v)

m5t phlng [P]:.{ + 3/ *

,[1(2; 2, 4) v;;

cIt

drrcrng tirana Ia)

I

l)-il
egiirjitu'sn.a
i; : .- . L-

')

b)d

;

.

tLt

'

-

I.)L

.1

HT

123.

:'-.17 t ): -'2 -. !, .lr' :.1.

,

r-1_
iP):2r-ry- z -r-1:0, I'I(1:2:-1). ,.Si A 2*9-5lt-2

,)

3.:

- 2 : 0,,11(3;-2:-1;.

Trong kh6ng gian voi hd toa d.6 7xyz, cho

.Br2.9.li) .Goi t.f-iAh)nhchieucr-ia,{
nin'r trong mrit'ph5ng
rT

v-i1

.

-'t

I

', ':)
:::_

-)

j'

i

(o)

dbng tho'i

I

DS:

mit phing

z*l

,'_)

.\

-6

(cr)

.:l.r -2u

- : -21) -

0

va hai dr6m ir 1: l:tj)

vi B tr6n (rr) .Tin}rd6dAidoan l:F.Timphr-rongtrinhduongthing J
cua \D vcti to r,') J vLrdng g6c vdr ,{8.

di qrra giao didm

;t

t r+

HT

124.
,Pr:.i:

Trong khOng gian vcri

-

2tt

hO

toa dd Oxyz, cno 2 m5t phing (Pl,

.:II:!
-z:0. {?, .i: -3ri-32--1:0. (dt

:+:

tal v) duong thang [d]

t,

'

IAn

luot c6 phircrng ti'inhr

tarpphLrongtrinhduongth5ng-\nimtrot.rg

[P] song song vdi m5t phing (Ql v) cit duong thang [C].

BE HOC VO

BCT

- CHUYEN CAN SE DEN BEN

Page 32

tt

-2-')
HT

I

125. Trong khOng gran

v0i

hOtoad6Oxyz,cho3di6m'-lil:2:-l)'13i'21'l;'CrU:l');v)dwongth-ng

*"t
:-:-: - ') :
LAp phuong trinh ducrng thing
2
2
-i
=,
pheng [ABC] v] r'uong g6c voi duong thang [dl
.r-2 lt-I :-.1
D /s: -t
2
t2
-11
11

A dr qua truc tam cua tam grac ABC' nam trong n'lit

rnit phing
iuong:hang I ::
HT 126. Trong khong gian vcri h6 toa dc oxyz cho dtam .ii.),: -1.1
-::
!p)..r tt .:z-it:0.Vidtphu'crngtrinhcuaduongthangd d1 quadremA namtrong(Plu;iop'diducrngthang

f

m6t g6c

Dls:

1:rir

Hoac d.

d:

, .t-:l

HT

u-2

727. Trong kh6ng gian toa db O:*yz, cho dlrons thang d: .)
\ri6r phutrng trinh dirnng thing -\
:!: *,t *;:
='2: t) Goi iv1 li giao di6rn cua d vi IPl

|

r

vudng g6c vcri d d6ng thoi khoAng c6ch tir M t6'i

l

-

a!

nanr r:()ng

mlt

Phang

(PJ,

bing jT
l'. I : -:)

:1

1

HT12B,.frongkhonggrarlv&ihetoac10o,t1,zchotnitphangirJ:,.,:

I

,

[rt]

,

t''

'

':

l1
r'i i-r) ch6o nilar-r u.ra khoing cAch gt'"ra thl"'g l'o''g
t..._.

l'-''

hodc rl ,lri:

-i

.

[::-,
trinh dtro

3rVirit
HT

129.

thing Ii6n uan

thi

kh6c

g6c chung cua hai duong thang
Trong kir6ng gian vor h0 toa dd Oxyz, vidt phuong trinh {wcrng vu6ng

Ir:3=ii
I

.\,

al6n hai ducrng

I

-1

.,:
\ ...l,
L,
1
vd J.)
- .
-l

,/

l::1-:rt

D/st

HT130.

Trongkhonggianvoi h0toa ddAxyz,vidtphuongtrinhduongthingddiquadidm 'iIi--l: -5:

[..r..,.1..=tr_n
r_!_tr_i
-..
::v) tr,t : ;
:
; r,

haidrrcrng thing: ,1, 1;'- .,;'=
["

tI i - -= 'nr,
: -5 l:I -;J-i

=

'll vicltca

:_L

=

I

D/s:
' d: 1l
)-

2t

i nffnl pflnt

Pase 33


-i,E

f

."

-

I.

't._: -i-i
-1-

,r, ];

D5

2t

l-

[.r'=3-21

b)M(3;la,

HT131.

1),Li

+:+

DS:

Trongkh6nggianv&i h6toad6 Oxyz.,cho hai dudngthing

i, _ )_,

r-r '-r
r-r*
t-t-+
112

l'-i.,-,-;-,J;. \
I-2: i
l:

f,

+,,.,#:3:=

giao didm cua

-\, vor I a I d6ng thrri cat

ti.)
D/s: r. l
I,- .LI

--,
1.

r!:]:_],,:,r'
l:=; zt

A..Arr.'im5t phingInJc6phuongtrinhli

n;:.7-,; +;-'2:tt.

Vi6'tphLrongtrinhdLrongthSngddi qtta

r'a vu6Irg gric v&i truc Oy,

rr

ii,t r:1 Trong khdng glan vor h01oE iolx-;'z cio iiu'r:ng th;ing 11, I : ,- 2/, clLrong thang t1. l) giao tuy6n.u,

l; - i-2i
hat :rat phSng [P]: 2r' ti 1 - t t'a [Q]: l: _ r.'- l: 1,- ''t Go?. li giao clidnr cita d, .rl' Vidt phrrcrng trinh
r

dLrcrng

thing d, qua diem A{2 3; 1J, dong

Lllt cin dinh
HT

thryi

r,7,

.rl, IAn lucr3 taflB va C sao c}to tam glac

D/s:ri . {r :2: lt -='.i.:::1.-'2i

L

'I'rong kirong gian vni he toa ci6 Oxyz,

133.

cit hai dr-rong thing

611o

nrat ph)ng

[): 1.: :] - i l: - ll

va hal ciucrng thing d1:

---

=

l-,ll - ----1, 1,-1 = - = a:r.Chu'ngnrinhring:l .,,'aJ:cheo.:m;ii 'r'ierphu'o'ngtr)nhdirirngth5rglnarl
:1.1.12
trin IPJ dring rhot \ c-it c:r ri: va dz.
D/s:-l. . - =
;)
--i
-l
HT 134. Trong kh6ng gian vcri hO toa d6 Oxyz, cho hai mit phing v) har dudng thing c6 phucrng trinh [P):
l--=r-t:--]
(d.) -:J-':-j=t-:
.1\-:l -.1.?. , ,t y2[Q1: .ir-J,-ttz-,:,.[dr]:
!1..i':;i1
Viet phirong trinh drrong thang [-\] song song v&i hai mat phing (Pl, ta) vi cat (drJ, [dzJ
ii - {'
l::,-1t,,-26:
D/s: [,\). -l- _:j-- _
lr :,
i+r
i

HT

135.

Trong kh6ng gian voi hO toa d6 Ox1,2, 96. mAt ohing [P]: 2:r: ' lj

:22 - 3 : 0 vi

hai dwong rhirng (d:), (d:)

VIC'Iphrrongtr)nhcluongthlng(\l
lAnlrrotcophuong1rin1','--}:
+-t
"
-:=-va+'l,l
:
:
:
3
-1
song songvoi mit pheng [PJ, cdt (ir,) vA (d") taiAvi B sao cho AB = 3.
._.)

,-1

-l

122
HT

136.

'

I
I

L

Trong kh6ng gian v61 h6 toa dd Oxyz, cho

rr-L r
,

"

g6c

.?
,,t
.
r
: J-j---'=
j "':
1

vdi a. vi cit

,C,

tai didm

,,-:
'i-1-s---':,
z

E c6

bD rrn

._)
-

mit phing [PJ: 2x -.y =;

=

1: 0 vi hai du]ng thang

" vi€itphtrongirinhdLrongthSng-\so)lgsongvoi

[P).r.'rrdng

-'llcLPrru

ho)nh d6 bdng 3.
rr^ Dry

arrrra/crr r

a

rr cE n#rr odrr

Dtoo

2A-


| .,,

Dls:A.

-

1:

T t.

-a

:6-t
r+S :,,j -6 :

Trong kh6ng gian vdi hC toa d6 1xyz, cho hai duong thing

HT 137.

,t:t
y- 2- t

.

'2

1

Vi6t phuong tr)nh duong theng (dl song song v6i truc Ox va cat (dt) tar A, Cal

[Cz)

tai

B, Tinh

, l * t/
-

t.
D/s

HT

-i;2
l''- -16

d

Y:
lJ -

138.

\.11p1

:12

Trong kh6ng gian vcli h0 toa
pl1yn.;1q

trtlh

ir
l,I
Ir
l"
l,='ii,
HT

139"l'rong

D/s A:

dlr i,-ng tirar-,g

c16

lhing: [d1l:

Oxyz, chtt har ducrng

idl s9n31:i:lrg ;ir;

r.l

- *i
-_il) - li
2.1

va

y*!
{-d2j:

)

')

1

:t

kh0ng gtan

rzcrt

h6 toa cio l')'i1'7' cho cic diem Af2'{J'ii); B[0'a'0J; cl-2'4'6;
\

tt

Viet pirrrcrng tt'rnh dtrdng thang A / /
! -l1-" ,

Iri: -',tu --- -]- -r,

j

ld) vA cat

cAc

drrong thing AB,

r'a d,-rnng

thane

ldl:

OC.

lJ, .j'; ',-('

)

Trong kh6ng gian vcri h0 truc roa 116 Lr.xl,z, cho b0'n didrn A[.1;5;61; B[0r0;1); C[0;2;0J; D[3;0;0J ChLrng minh
goc vcri nrrr p]rang O'x-v va cAt cic
cAc dLrdng th;ing AB v) CD ch6o nhaLr. Vidt phirong trinh dti'irng tllSng fci

140.

"'r-tirng

du'ong thAng AB, C).

HT

"rr/ -

r.ii il:l r.'.) ,:ii ca i';ll dti'rvrirl tlSng i.i7'r, . r-,

l:="

J.

Irr'"
l''t-:lt

HT

:

141. Trong kh6ng

gtan

vtri n0 toa d0 O,tyz, cho h:ri

tlr'rtrng thing c6 phucrng trinh:

lr

: -r - zr

,1.. ri;::

vi

]' --,-,
I

d,
cit

trungvdi gdctoad6O,
_.Xetu,t+i,tuongd6i ciadrvad:,ViStphuongtrinhduongthAngdquaivl

I:;-

dr va vu6ng g6c vdi d:.

I,.:'
I

D/s't lr: -,

l':

I

CAu

)

hdi twdng tV;
1,,

a) VOr )i;1.7.L;

,

I
1

*I

_ ZL

I

i!l.j

-i:tt
2+t

ES; ,l

:r'-1
:

31

L1-l

l


1.,:

t

I

va

[d:]:1!:l-t

Trongkhong gian voi h0 toa do Oxyz, cno 2 dLiong thing c6 phucrng rrinh:

HT 742.

(drJ

I

lz:0-2t.

L_,
-.
l,

jir:31'-i,.Goi KI)

t'-

l':

-i;11

hinh chi6u vu6ng g6c cua di6'rn

tr6n {dzJ. Tim phucrng trinh tham s6 cua du}ng

r

thiing di qua

vu6ng goc vdr (d1l

K

lx

vi cit (drl

I

11\

11

(ri

12

)

"
l

--

Lt

,],

\

-,)\J/\

.-1-t.l
11
HT

143. Trong kh6ng gian voi h0 toa d0 0;;lz chc drdm [0;1;1] va 2 ducrng thang. [dii [d;] voi; (cirl:
't - i: li -') :i ;
]i giao tuven ciia I r.et'-rl-,anr t.P., - i : rl ,.,a IQJ : :' , ,; : -- ) - li. \/i6t phLrcrng
iv1

.)

I

i

[c]rJ

trinh du'ong thing [dl qua

D/sAM: i

M vLi6ng 96c

rd-]
[d )'"a ie:

'i't-L=,=]

.):

HT

744. Trong kh6ng gian ;cvi h0 ioa ii.
.i--i

U- l

;i

|
..P-1

I

:

. ] -)
D/s --r : --'
-- -

.

'l

vLrong goc \'(ir duong

HT145.

t-

.C.:.,r

)

^+,.r^-.1,,,;.,.-rLinor\,.i^.irnn.rr,rirrhi

*rhing

r,i1 va cat JLr org.

l: : 1; \r) 2 dt|crng tirars

z, :ho :rat pirang r^ I

:ra:e :',

-

Trong khOng gian vcri h0 toa dd Oxyz, cho n-.al pha;rg rP]:

lx - -'

,..1::11
iPl, vu6ng g6c vdi drrd'ng thang [dr) vA cat dLr ]ng th;.ing i.i-'1 irr d,enr

E ': ' r:nn

,

: ir

C0 bang

12 h-r

':'Lr

'.'nr :hang [d:]:

3

fr-ll-t
I

Dlsr!:].-;
l.'-tr
t.
HT

146. Trong kir6ng gian vdi hO toa do Oxyz, clto hai didm A[0, 0;-3], B[2, 0;-1) vi mat pheng [P) co phrro'ng tr)nh:
q-' 81,, :7;. * 1:0 ViSt phuo'ngtrinh cninh tic d.rlng thing d nam trOn n-,at phang (P] r'i d vtiong goc vtri AB
tar giao drem cua drrong thing AB vo'i IP)

, -)

.:

'
HT't+7.
'I

,,
-!

---1

l

Trong khong gian vdi h€ toa d.o oxyz,cho har cluong thing"

- 1:
,12

y

- 2 :' - 1 ua mit phing [pl: r - !/ -

2:

-;] :

t) ,

dr j--L : +:::f
)
-L

;

t

dr:

vi6t phuong trinh duong thang -\ nin.r tr0n mit

phang IPJ vA cit hai duong thdng d1 , d:

,

t-'.
D/s:A:' -':I:
i
3 -i

HT 148. Trong kh6ng gian vdt h0 toa dO Oxyz, viet phLrong trinh duong

th5ng [d]

g5c r,6i

mit phing

1/

i-,,-:-

i
1

.r:::D/s

I

l-ndbngrhcyi catcahai du'ongthang ,;

)
==

":- - I: va
-I

i

-l

,VC/I irll

I

od.r... rrA piv rrrrrtrilr rhr.r ci niu pf,u

PasP 16

(Pl


.1

Lt, .I --

5

tvang tV:

Cdu h6i

a) ltni (P): -lx

- J - iz - 3 :

A,

:L:*l

trt.

P

t

-7
')

2

" 1

" ')

z*2

215
b) Vai

'l'1:2.t: t1 i: - 1 :

tl

, d,

f-t

y-l

.

3

2

i

DS,
HT

149.

, :r-2
tI'r-2
''r,i1.:,1
I

!l-2

1

Trcng kh6ng gian v&i hCAoE dcfDxt,z, cho ba mat pheng: [PJ:

z

-11

2r !; i -,:

, [QJ:

Lr

.!' q -

2.:

- l]:

lr,

.'1:*i-tlvaJu'ongthingA, i_:l:!:]::
Cqi 1,, ligiaotuye,cua(plvatQl.VietpSuong
ll:.i
r.i-rr'li lirrrrttll thin;1 [r1] viri,:tg gr-rc',,ri ilil ,;:.r ,:il ii nai iirrirttq t)ring -.:, . -i,
[R] :

D/s

HT

.r-'),:1

i
,_

d:

1

2;l

.-

r

'l

I

150. Trong khOng sian vfvi h6CIto8 d6E Ox-r'e, ciro ba drrdng thingc6 phrrcrng
''2
,l

,lt

L

,.,i...

D/s:

ti.,

__,1

' '':t

i,

2

"-- r"-'-"D -'-"''

I

lAnluottai cacdidmA,B,Csaocho,;1 R

- U('.

\

Dang 4: Vi6t phucrng

trinh duong thing lien quan d6n khoing

cdch

ir -IHT

151.

'l'rong 1
cho dLrong rSng [dJ: I
r

I

l^

-.t - .L/ r 2: * 5 :
1A

trinh

Jil

i'
.)

:',
)

l. - -r (l . Vi6t phucrng

trinh du'dng th;ng [-\J nim trong

[PJ, song song

va mir phing

i,6i [dJ vi cdch [d) m6t khoAng

.

:=i:
l

. -J

hnna

:--.

.

-1.)l

-

HT

(P)

l
'

152. Trong khong gian voi hO toa do 0x),2, cho mit ph5ng (P): : - j/ : - I 't -2 : !-] : t -, t,rn, Ilagiaodidnicr,rar?vA
[PJ.Vietphtrcrngtr)nhcuadtrongthang
1
3
-l
g6c vcri d sao cho khoang cach tir I c16n A bing 11 : 3r 2
,

-1
Cdu hdi tLrctrtg tir:

0

vi

ducrng

thing:

d:

A nam trong [P], vu6ne


a) (.P'; t- :

},

: z:

tJ

*)

.rl
_

ll

l
-,

,l

HT153. Trong kh6ng gian voi h6 toa d0 Oxyz, cho mat phing [PJ: 2x-U-2t=i):0 vi duong thing
p-1 -':-3 vietphuongtrinhduongthingAvudngg6cv&i
r-i
[FJvacitdtaimotdidmMcich[P]
' I
:
(l-

-l

m6t khoing bing

It

2.

ro

7

' - __ _

iL=-

)+

1i

11

2r)

D/s:l
' 1,,:-l=,
11
ll11
_ _.)f
It" - iI

154.

t

'-]
^
^

l-'
.) (t
-t1

-

,tf
-l

trinh dulng thing li6n quan tl6n g6c

Dang 5: Vidt phuong
HT

hoac):

Trong khdng gian voi h6 toa d6 Oxyz, cho di6m A[3; -1;

1) duong thlng A:

'

tr

-2

,. ;

..
,i
- phing
- , ui m5t

[P]: :r-lJ+z-5:0.Vi6tphucrngtrinhthams6criadwongthingddiquaA,nimtrong[P]uiihqpvtii cluirng
thing 1 mot g6c 1.-)lr
.

t.,

,)

,

D/s:.i:

l

-

i.'ltoacd:
I

l:i HT

I

155. i:'ur.;
iP):

.r

*y

ki:

jrrg

z+ 1:

h6 toa d.6 Oxyz, vidt pliLro'ng trinr dil'ong thang.\ nirn'.rcng mat

r',- l-i
.
r',1 'L.lO \,'t . .,
;'. ...:
If,C'. JI.\
. r/.
0, cdt c6c dulng thing d,
,
.11-l
: - -J
ri
:= )'2i
grarr

phang

l

I

i

f,:i+r

[r:i

D/s:d:lv: -t hoacd: ju: -t -t
[z=5+,
lr:5*!
HT 156, Trong kh6ng gian vdi he toa do Oxyz, cho h)nh cirtip A.0BC, trong dri A(1.;2; -ll, B thu6c truc 0x va cri hoinh

d6 dtrcrng, C thuoc Oy va co tung d6 drrcrng. Mat phdng [ABCJ vuong 3crc uin mat phing (OBC1, urrdit'- ].lrieit
r

pirucrng tr)nh tham sd cria duong thdng BC

HT

157. Trong kh6ng gian voi hC tqa d0

Ox1'2,

cho hai cliem

-{r'?r

-

1:I j.,910:

1: 2) v)

dr.to'ng

e:{:a-3:a::Vidtphuongtrinhducrngthang-\cli quagiacdidmcliadrtongthingdvcvi
1 -r
I
[0AB], nim trong mit phing (OABJ va hop vdi du']ng thing d m6t g6c cr sao cho .,r. o : l

rhang

rnat phing

6

D/s: J:

r-10

2

t1

-13

-5

::21 r-!oac
, - ); .r-1i)

-11

6

li

-i,l
l

:-11
-i

HT15B, Trongkhdnggianvoi h€toaddOxl,z,vi6tphucrngtrinhdr-rcrngthingldi quadidm
RF HOT l,/N RTY CHII\/FN CAN qF'r)EN IIF'N

l(i):

l: -2,,r,u6ngg6c
Peoe ?R


v6idrrongth6ng rJ

:r

:l:3:a
_:l
i

l'.

i'='
rl'E/s:-4'Jt-t-thoac
l,-

HT

159.

a:

o

Dang 5: Vi6t phucrng

I

-i

lu:)
l,-

uataovdimitphang[P): 2x r'!-z-:.5:f)

i

-)-')i

trinh dudng thnng li6n quan tl6n tam gi6c

v) phuong trinh dulng cao AH'
Trong kh6ng gian v6i h6 toa d6 Oxyz, cho l-4 BC v&i toa d6 dinh c(3; z;31

r-l
d' l:5

r-2 u-:3 :-:l
:=='::=

f .1l]C vi tinh diQn tich crla A.4BC'
=+ P) -rlr'=>"P;: 't:i y -22+1:O

phrrong tr)nh ducrng thing chrla canh BC c0a

Gqimp(P)qttaCvdvudrtgg6cvaiAH

3:tP)a'{l ),+
Goi

mp(Q

Bll;4: 3)

=phuangtrinh

=+

Lap

'

)t:tt :}-2t"2:3

'i':1.-

i{i2:2:1)=+ llt1"2'5) {Kldtrung didmcfiaCM)'

-.1
160. 'Long
phil'on1_.q

BC

---:l
Y'+ - ----

qua C, vu\ng g6c vdi 6u (Q) cdt dz va AB tai K vd M' Ta c6:

(Q) r -2u + z-2:0

HT

a:300.

'

phLrcrngtritrhduongphingiActrongBD )An lLrcvtl): rl,

.

m6tg6c

l'lnn

l:

)'.-t

-....,r,,.

i

la

tu-

I

!

i, "I

:;l i, lTii : r.,,

J-l l:i(' v0\

.l(l:

' l: l) vi

hai dwong trung tuydn thn lucrt

I

I

i/.:ir'-r'
j:=.J'i

h

rro'

I

nfirr

Peoo ?Q

co


'

vIfT pHrroNG TRiNH rvrAr cAu
HT161. Trongkh6nggianvdi h0toad6 Oxyz,chodr6m lL; -2:
)
Oy. E/s: (:r: _ i,2 -iy *212 - (: - 3 j2 : l0
HT

162.

r
ri, :

Trong kh6ng gian vdi

f,:zt
[r:
: t vA d, : :
:

[.

l.:

vu6ng g6c chung

D

/s:

(t:

Cdtt hot

2'i'

-

-

r/r.

d,

trinh mit cAu ISJ c6 duong

ch6o nhau. ViSt phuong

dr. rti.

-

-

l')2

-

:i

i'

:

2;!

n]r

l)

doan

1.

rlrarla tv:
l-

-2

.r

d.,

tt

-

1

-1

0,,1;
2

'2

-r

O

+r'

:;,-''
-

DS:

l.-+]' *[r*fJ' -f,-il'

(s)

[' :'

- lJ*l _:

l::-2

',

(dz)
2

.)

y-4

1

163. Trong kh6ng gian vryi hA tQa d6
: :

0

2

(r

,

-2):'

,'2

+i,{ - -il') -,, - ,t,'
l

, -

5

z-'2

DS: (,e) ,

HT

kf

i,

I

a)

minh

. Chr?ng

cia

(.u

.

i

t

la

r

lA

.Vi6tphuongtrinhmitclutAmlvltidpxricvdi truc

toa d6 Oxyz, cho hai dLrdng thing:

hO

:-

:3;

ax1,:r,

cho hai dwcng thang:

:l l
,1, ,

'Il

ll -- i - ,- l - --,i

Viet phutrng trinh nrat ciu co bin kinir :riro nhit tiSp xLic vcli ca hai rilrrrng trang

rr va

va

r.,.,.

= doa, vuang g6c chung cia hat dvitrct thang la clu,on,q kinh.
'.'.iat cau nhdn
Cdu hdi tvono tu:

i ,-- .i -'r*
i -),!t :t
a) ,i.-l-_,-l-_
, i., ),1 : t

[,.

[r=*
HT164,

r

,5.

1.t: : i.r:

. 2ll -

ti

.- ltj - l.r - 1,: :

I

[e=0

Trongkhdnggian r,&i

ij, i cri phuongtrinh {.2, :2i ,1/: t:z: L; (-rr; ia
graotr-rydncua2matpiring ,r :i *lt- 3: l, vi r..i;:-lr 1,1
-3:- i2: U.Chrlngt6hartlu,ongthing J,. J,
ch6o nhau vA viSt phLrong trinh rnAt clr-r nhAn doan vu6ng goc chung cria
-\. J, lAm duong kinh.
D/s: .- l,-- r,,-L,l -: lr--

HT

165.
v6i

I

toa do Oxyz,clto d..r'ong tlrang

Trong kh6ng gian v&i h6 toa d6 Oxyz, cho dr6m A1L;

:

?

hO

+

:

f

,'"n

-Z: 3J vi duong thang d co

khoing cach tt diem A ddn duong thing

d.

phLrong trinh

vidt phuong trinh mit ciu tim

A, tidp xiic

d.

D/s: (r - ir2 - ,,tj -21- i.; - 3ll :

50

i
I

i
i

HT 166,

Trong kh6ng gian vdi h6 toa dd Oxyz, cho duong

thing

- * i
at

I

i

tnan d cit

mit c)u (!J,

c6 tAm M, tai hai didm A, B sao

I>ti rrnr

rrA niy

n:rrrrrfirr

cho AR
n'i.'

:

cF

6

:

. :

vidr

,,jt

-

_:

: : v) didm .11 i: t:ri,,
trinh cia mit ciu

Duong

I

ISJ.

I


D/s:

fSJ:

-U-\2 -(z- (:i!:ts.

{.c*7)2

Trong kh6ng gian v6i hC toA d6 Oxyz, cho mAt phdng (d .2r * y - 2: - ,l : 0 vi mat c'Au
(S'),.r2 o !t) + ,2 -'2., [!i 8; .1 - L]. Xet vl tri tr-rcrng d6i cira mit chu [S) v) mat pheng
'.,.). Vi6t phuong
- - -

HT 167.

trinh mat chu [S'J ddi xung vsi m5t ciu [S) qua mAt phing i1i '

E/s: (S'l : i:r - 3.i
HT

HT

-

g)

-:- ::.1

.

:2it.

168. Trong kh6ng gian rr6i hA toa do Oxyz,)Ap phuong trinh mat chu [S] biSt ring mit
[P]: : : 2 lhn luot cit ISJ theo hai dr.rong tron cd bdn kinh blng 2 vi B.
D/* (S): it: - o1! -:y - tt,2 -,r - 1U,1 : 261) (a, b e R).

phing Oxy vi mit phang

169. Trong khdng gian vdi hO toa do Oxyz, cYLo mit phing [P]: 2.,: - rt - 2: I - U v) duong thang d:
I 'u'ri t-l.Vietphrrcrngtnnhmatcau [S) cirtAm lt].rr-r6cd, IcAch [P] m6tkhodngbing2vA [P] cit[SJ
*) - 2 1

theo nrdt dr.rong tron [CJ crj bAn kfnh bing

3.

a'

H't

170.

Tror:g khdng gian vri'i h6 toa dd Oxyz, ch) Z dieim,\i0; tl; al, Bf 2; 0; 0l

LAp pirLrgng tr)nh nrat cAir ISJ di qLra O, r\, B r.a co l
vi m;t

cich t,* tArr I lira ln;t c:u

pheng (Pi: 2x
cld-n

-

N

-- :

-'ir

:rrat pir5rrg (1)l i:alrg

'--

(l

:
.!{

H'f

171. 'frong }

,.r)rr: ,'2.11 -;'2: I-(t .Lipphuor.rgtr)nhmatcAu(Slci'inir

drOn tich hinir chieu cua tam giAc

D/s: i.:' : i.r - i,: -

HT

t7?.

l.r1

'-ltl - l: - ill

.'{Nt' tren mat ph.lng t,
'=

l;

-\

i-

1rh

I

lri

d:
Trong kh6ng gian voi h0 toa do Ox1,z, cho dulng thing
-31

t

t
. : +)

:;

va m5t phdng

[P):

2r-:,rr*)t.-2:0.LApphr-rongtrinhmitcAu[SJcritAn.rnarltr6nducrngthingdc6binkinhnh6nhAttielpxirc
**
J
vi
di qua didrn A[1; -1; 1J.
[PJ
D/s: (:r - 1): * (v - t;2 - :2 : 1.

vdi

'
HT173. Trongkh6lggranQx;,z,choduongthinsd:
)..."...b..

1-''-'2::v)matpheng[P]:

2t:-

11

vdi rnat pheng [P] v) di'qua clidm A[2; -1;

0J.

2:-)-'l

.Lip

1

phucrng trinh
D1s: l,{1

: i.1

mit
2':2

cAu [S] c6 tAm nAm trOn d, ti6p xirc

-',t1-'L)l - t.-

,'] 201
I
hoaclS): I.r--l

[

HT174.

13

i

1)2

.

=i'-*j'

:

1

i ..,.1;l

,l

IJ i

l2l
1rl:r

Trongkhongglan volhOtoad6O.xyz,chodidrr 1ii: !; -2) ,ducrngthingl:

2:r'-2: i/-3-:

r'imitphing

citkh6i chutheothi6tdi6n

[p]; 2r'n2r,*z:5:0.\rr6tphLrongtrinhmitciu[sJc6tamlsaochomdtphAng[Pi
l) h)nh trdn co.chu vi bdng S; , Tu do lAp phuong tr)nh mit phAng IQJ chria a vA il5p xiic r'6i [S].
Dis: lS) ;i.r'-L'r2 :iy-'2,\2 -(:--- )l -2|: Q).6,- -331-30: -10i:0.