Bài tập hình giải tích trong không gian 20162017

ai lich bans j Dang 1: Vi6t phuong trinh clucrng thing bing c6ch xic tlinh vects chi phuong HT 114.. Trong kh6ng gian vdi hC toa d6 Oxyz, vidt phrrong trinh hinh chi6u vu6ng g6c cua dLro

I' - 'tiww &,oeL ?"*tt"5

I

t vtrr PHLPgNG rniNn ruAr PHit'tc

D4ng 1r Ccr bin

- -?,,-)- -5:0.Vl6t

t J, \ a4

,Qt,2A-3;-11:0.

vi song song voi duong thing a :

vsl h6 toa do Orvz , cho hal di€jm AQ;a \' B[-1;1;3J va mat oh'Ang [P):

T

v

z

Dls:(Pl ii.c- j1, 'r- -9:Lr'

HT 88 Trong khdng gian vcYi h0 toa

.:) .- !-l) - .;l - ?x - 2,' *'\z * 2 : 0

xirc vdi mit clu [S1.

,,., I :+ :1.j Chfrngminh ringdidm i1 d.,.,?,

-1r:J

phang d6 D/s: t: r2Y -.2 r ? : 0 '

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

lr-rit _i _ +

-')r

tQa d6 Oryz, cho 2 du&ng thing (4) vir (d.1c6 phuong trinh:

'-1 :! -,\ := Lip phuong trinh mit phing IPJ chria [d,) vi (dr) '

693

HT 86 Trong kh6ng gian voi hC tqa dd Orqz , cho mat ciu tS) c6 Phuong trinh:

song song vor gia cua vec rry : : , tl::

,' - !t't + z2 *2t+6y 12- 2:A Viert phrrong tr)nh m5t nhang (P)

ru6nggocvmmitphang 'r i - l, -: - II L vAtiepxircvdi (S '

t-''

.d''

D/s:{P): Y+Y-52+10=0

:r, "!_3

dO Oru: , cho dr-rong t'hing d: t;:

T

LAp phr-rong trinh mat ph;ng (PJ song song v&i d vi

trLLC 0x, dbng thcn tieP

HT89 Trongkhdnggianvo,rh0toa dO Oru:,chomitciu[SJ: rr -!2-t2-2''-1';-]:0 vAmitphing[P]

'1,Ii3:i: -1) vuonggocvoi mitphing [PJ vatrep x'ic

vryi mat ciu [Si.

Cdu hoi tvqno tLr:

yri (Sf , 12 = y2 + :2' -2r - L'J - iz i5 : 0' 1P' : 2r - I -6'-5 : C''11(l:1;2)'

HT90 Trongkh6nggianv6i hCtoa dd Orlqz,chomitcau[S]: 12 =92 -'2-2'*1'l*22 -3:0 Vi6tphugng

irinh mit phdng toi.no, truc ox r,) cit mit c;u [S] theo m6t dudng trdn c6 bin kinh r : 3 '

Dl* Q):Y - 2z = 0.

HT91 Trongkhonggianv6ih6toa dd oiy:,cb.omitclu[S]: 12 -!'2 -:2 +2r-2y-22*7:0 viduongthing

-;lr 'ffiTil.frur ' \ it - r - 6: u tS) &fu" ,t *taivz w

BJ s* l{* :t-e \1* ). * !,.

HT113 Trongkh6nggianroa do O:tu:: chocacdi€:n l(ll ll: ir;.Bir:2: i) Vi6tphuongtrinhmltpheng[P]quaA,B

/

DS: lP;: .r *211 - )z- 3 : 0.

ai

lich bans j

Dang 1: Vi6t phuong trinh clucrng thing bing c6ch xic tlinh vects chi phuong

HT 114 Trong kh6ng gian vdi h€ toa dO Oxyz cho cluong thang ,l ' ; t :

,i l) : - i : t) Vrdt phuong trinh duong thang -\ di qua -111: 1: -2 , song song voi mat phing

vdr drrlng thing d

, L,_1 :',)

HT 115 Ti'ong kh6ng gian viri h6 toa d6 Oxyz, cho

: - l -:- t [t = ,q] vi m5t p]ring [PJ: 2.r - u -2: -:]

[P], cat vi vu6ng g6c v&i (dJ.

l.i

I

D/s: J j,r7 : -;J

I

l:: t - ' .

dudng thing [d) c6 phlrnng lr:hl: { - - i : tr : -1 -'2t ,

,,

Lip

Gqi (U la mdt phdng qua A, B va vulng goc v1i (P) = (U, Bx + 7x + i1'z - 46 = 0

iD) = (P) ',

Q) suy ra phLrcrng trinh {D)

HT 118 Trong kh6ng gian vdi hC toa d6 Oxyz, vidt phrrong trinh hinh chi6u vu6ng g6c cua dLrong thing

1 n- t\

[" t'' ] -'i-t

E f s:1:

Cdu hoi tvong tLr

t.t.:++tllt

t.

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

l : i- tot

t

[':

I

1!t :

t^

I"

.r -1_ r-1

r

-.)'1,r,

2 29rrt

) - Jltt,

HT 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.

Bd Hoc v0 Bg - cHUyEN cAtt sE oEN sEI{ Page 31

l": !-

Ir

:i -: Cl

l:t

D/s: d: ].,, : : - Zt

t')

I

L -r - I - rlr I

IIT 120 Trong kh6ng gian vdi h6 toa d6 ?xyz, cho 3 didm A(L:2:-\).8(2:l:1;:C(():1:2i v) dudng thing

, r-l u-l '-)

2 -',_ 2 phing IABCJ vir vuong g6c vdi dudng thing d.

' ') -l ; -1

Dang 2: Vi6t phuong trinh dudng thing Ii6n quan il6n m6t Clucrng thing khic

HT121 Trong kh6ng gian v6i hC toa d6 Oxyz, cho didm M[2; 1; 0J vi du]ng thing d c6 phuong trinh

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

D/s:J ' '-j= t :t -l | ' -l l/'ll -: l,l ;i -1,;j'

;t22

2z r 2 - 0 l-Ap phiro'ng tnnh clLl'rrng rhang,l song song vdi nr5t phing [P) dr qua ,[1(2; 2, 4) v;; cIt drrcrng tirana Ia)

,- ,, ) - ') - - I

l)-il

egiirjitu'sn.a tLt

.

i; : - L- - ' j' :'-.17 t ): -'2 - !, lr' :.1 .

I

.1

,) 3.: - 2 : 0,,11(3;-2:-1; DS: .\ ,'_)

-6

HT 123 Trong kh6ng gian voi hd toa d.6 7xyz, cho mit phing (cr) .:l.r -2u - : -21) - 0 va hai dr6m ir 1: l:tj)

nin'r trong mrit'ph5ng (o) dbng tho'i I di qrra giao didm cua \D vcti to r,') J vLrdng g6c vdr ,{8

t r+

HT 124 Trong khOng gian vcri hO toa dd Oxyz, cno 2 m5t phing (Pl, tal v) duong thang [d] IAn luot c6 phircrng ti'inhr

,Pr:.i: - 2tt -z:0 {?, .i: -3ri-32 1:0 (dt .:II :! :+: t, ' tarpphLrongtrinhduongth5ng-\nimtrot.rg

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

r-1_ lt-2 z*l 2*9-5

; -) :::_ ', ':)

I.)L

')

-2

HT 125 Trong khOng gran hOtoad6Oxyz,cho3di6m'-lil:2:-l)'13i'21'l;'CrU:l');v)dwongth-ng

11 :-:-: - ')

*"t

:

=,

pheng [ABC] v] r'uong g6c voi duong thang [dl

.r-2 lt-I :-.1

HT 126 Trong khong gian vcri h6 toa dc oxyz cho dtam .ii.),: -1.1 iuong:hang I ::

-::

!p) r tt .:z-it:0.Vidtphu'crngtrinhcuaduongthangd d1 quadremA namtrong(Plu;iop'diducrngthang

f m6t g6c 1:rir

I v0i

HT 727 Trong kh6ng gian toa db O:*yz, cho

:!: *,t *;: ='2: t) Goi iv1 li giao di6rn cua d vi

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

l

- a!

-\ nanr r:()ng mlt Phang (PJ,

r bing jT

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

t _.

l'-''

hodc rl ,lri: -i .

[::-,

l' I : -:)

:1 1

I

3rVirit

HT 129 Trong kir6ng gian vor

.\,

-1

I

trinh dtro thing Ii6n uan al6n hai ducrng thi kh6c

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

Ir:3=ii

I

-l

l::1-:rt

D/st

HT130 Trongkhonggianvoi h0toa ddAxyz,vidtphuongtrinhduongthingddiquadidm 'iIi l: -5: 'll vicltca

[ r ,.1 =tr_n

- t r_!_tr_i :_L haidrrcrng thing: ,1, 1;'- .,;'= ; : r, v) tr, : ; ::- =

=

["

t 'nr,

I i - -=

I

D/s: ' d: 1l )- : -5 - 2t

l: -;J-i

I

f "

-,r, ]; 't._ :

l-I.

-i-i -1- 2t

D5

[.r'=3-21

l:=; zt

HT131 Trongkh6nggianv&i h6toad6 Oxyz.,cho hai dudngthing A Arr.'im5t phingInJc6phuongtrinhli

i, _ )_,

l: : i

giao didm cua -\, vor I a I d6ng thrri cat 1 r'a vu6Irg gric v&i truc Oy,

ti.)

D/s: r l

I

I,- L- rr

ii- r

,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

Lllt cin dinh L D/s:ri {r :2: lt -='.i.:::1.-'2i

HT 133 'I'rong kirong gian vni he toa ci6 Oxyz, 611o nrat ph)ng [): 1.: :] - i l: - ll va hal ciucrng thing d1: - =

:1.1.12

trin IPJ dring rhot \ c-it c:r ri: va dz.

Viet phirong trinh drrong thang [-\] song song v&i hai mat phing (Pl, ta) vi cat (drJ, [dzJ

D/s: [,\)

-l-i+r _:j _ lr :,

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:)

song songvoi mit pheng [PJ, cdt (ir,) vA (d") taiAvi B sao cho AB = 3.

._.) ,-1 -l 122

HT 136 Trong kh6ng gian v61 h6 toa dd Oxyz, cho mit phing [PJ: 2x -.y =; = 1: 0 vi hai du]ng thang

[P).r.'rrdng

" ,

:

1 = j "': z - , -

-'llcLPrru

g6c vdi a vi cit ,C, tai didm E c6 ho)nh d6 bdng 3.

bD rrn - rr^ Dry arrrra/crr r a rr cE n#rr odrr Dtoo

2A-I

L

D/s d

- -a T t.

:6-t

,t:t

, - - l * t/

-i;2 :

-16 :12

1:

t.

l''-Y:

lJ

-r+S

'2

:,,j -6 :

1

A, Cal [Cz) tai B, Tinh

y*!

')

"r- 2.1 - *i r/ - -_il) - li va {-d2j:

\.11p1 pl1yn.;1q trtlh dlr i,-ng tirar-,g idl s9n31:i:lrg ;ir; r.l r.ii il:l r.'.) ,:ii ca i';ll dti'rvrirl tlSng i.i7'r, r-,

ir

l,- I

:t

Ir

l"

l,='-ii,

HT 139"l'rong kh0ng gtan rzcrt h6 toa cio l')'i1'7' cho cic diem Af2'{J'ii); B[0'a'0J; cl-2'4'6; r'a

I , Viet pirrrcrng tt'rnh dtrdng thang A / / ld) vA cat cAc drrong thing AB, OC.

l''t-:lt ! -l1-"

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

D/s A: )

lJ, j'; ',-('

HT 140 Trong kh6ng gian vcri h0

cAc dLrdng th;ing AB v) CD ch6o

du'ong thAng AB, C).

d,-rnng thane ldl: l

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

nhaLr Vidt phirong trinh dti'irng tllSng fci "'r-tirng goc vcri nrrr p]rang O'x-v va cAt cic

lr : -r - zr

]' ,-,

I

d,

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

cit dr va vu6ng g6c vdi d:.

I,.:'

I

D/s't lr: -,

I

l': )

CAu hdi twdng tV;

*I _ ZL

2+t

:r'-1 L1-l

31

1,,

I

i!

l.j

I

1

HT 742 Trongkhong gian voi h0 toa do Oxyz, cno 2 dLiong thing c6 phucrng rrinh:

lx

.r:::- 11\

11

" Lt

.-1-t.l 11

1.,: t

I

[d:]:1!:l-t va (drJ I

lz:0-2t.

I

{dzJ Tim phucrng trinh tham s6 cua du}ng i

1

I

g5c r,6i mit phing (Pl

1/

-l ,VC/I irll

tr6n

L_,

l, -.

l': t'- r

D/s (ri )

HT 143 Trong kh6ng gian voi h0 toa d0 0;;lz chc drdm iv1 [0;1;1]

't - i: li -') :i ; [c]rJ ]i giao tuven ciia I r.et'-rl-,anr t.P., - i :

trinh du'ong thing [dl qua M vLi6ng 96c [d )'"a ie: rd-]

.):

.i i U- l : I t- )

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

*- P-1 vLrong goc \'(ir duong rhing r,i1 va cat JLr org :ra:e :',

D/s r : ' -

, 1::11

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)

'I - 1 : y - 2 :' - 1 ua mit phing [pl: r - !/ - 2: -;] : t) , vi6t phuong trinh duong thang -\ nin.r tr0n mit

,12

phang IPJ vA cit hai duong thdng d1 , d:

,

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

i 3 -i

HT 148 Trong kh6ng gian vdt h0 toa dO Oxyz, viet phLrong trinh duong th5ng [d]

:- :

i-,,-:- l-ndbngrhcyi catcahai du'ongthang ,;

==

" - I va i

Lt, I 5

Cdu h6i tvang tV:

" 1 " ') z*2 215

tI'r-2

:L:*l 2

')

2

i

y-l

3

f-t

''r,i1.:,1

DS,

HT 149 Trcng kh6ng gian v&i hCAoE dcfDxt,z, cho ba mat pheng: [PJ: 2r !; i -,: Lr , [QJ: .!' q - 2.: - l]: lr,

ll:.i

r.i-rr'li lirrrrttll thin;1 [r1] viri,:tg gr-rc',,ri ilil ,;:.r ,:il ii nai iirrirttq t)ring -.:, -i,

D/s d:

HT 150 Trong khOng sian vfvi h6CIto8 d6E Ox-r'e, ciro ba drrdng thingc6 phrrcrng trinh

''2

D/s: \

Dang 4: Vi6t phucrng trinh duong thing lien quan d6n khoing cdch

HT 151 'l'rong 1<h6ng gian vo'i h6 toa d0 Oryz

-.t - .L/ r 2: * 5 : (l Vi6t phucrng trinh du'dng th;ng [-\J nim trong [PJ, song song i,6i [dJ vi cdch [d) m6t khoAng

ir -I- i'

I

l - -r - '

-J :

-1.)l

HT 152 Trong khong gian voi hO toa do 0x),2, cho mit ph5ng (P): : - j/ : - I - 0

't -2 : !-] : t -, t,rn,

Ilagiaodidnicr,rar?vA [PJ.Vietphtrcrngtr)nhcuadtrongthang A

g6c vcri d sao cho khoang cach tir I c16n A bing 11 : 3r 2 ,

:=i:

-

hnna l

Cdu hdi tLrctrtg tir:

-1

a) (.P'; t- : tJ : z: }, *) .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

' r-i p-1 -':-3 vietphuongtrinhduongthingAvudngg6cv&i [FJvacitdtaimotdidmMcich[P]

m6t khoing bing 2.

I ro

t ' - _ _ _ )+

i - 11

L=

D/s:l 1,,:-l=, ' l- 11 hoac):

l11

I -

- _ _.)f

t" iI

7

1i

2r)

t

'-] .) (tl-'

^

- t1 - -l

h6 toa d.6 Oxyz, vidt pliLro'ng

Dang 5: Vidt phuong trinh dulng thing li6n quan tl6n g6c

' tr - 2 - , i

HT 154 Trong khdng gian voi h6 toa d6 Oxyz, cho di6m A[3; -1; 1) duong thlng A:

, ; - , ui m5t phing [P]: :r-lJ+z-5:0.Vi6tphucrngtrinhthams6criadwongthingddiquaA,nimtrong[P]uiihqpvtii cluirng

thing 1 mot g6c 1.-)lr

t.,, ,)

I

l: - I

i

r',- l-i 11-l r/.

:= )'2i

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, r urrdit'- ].lrieit

pirucrng tr)nh tham sd cria duong thdng BC

l

, r',1 'L.lO \,'t ;' If,C' JI.\ J ., : : - - I

ri i

iP): .r * y z + 1 : 0, cdt c6c dulng thing d,

f,:i+r [r:i

D/s:d:lv: -t hoacd: ju: -t -t

[z=5+, lr:5*!

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 rnat

6

rhang

phing

RF HOT l,/N RTY - CHII\/FN CAN qF'r)EN IIF'N Peoe ?R

v6idrrongth6ng rJ :r _:l :l:3:a uataovdimitphang[P): 2x r'!-z-:.5:f) m6tg6c a:300.

i'=' l' '

Dang 5: Vi6t phucrng trinh dudng thnng li6n quan tl6n tam gi6c

HT 159 Trong kh6ng gian v6i h6 toa d6 Oxyz, cho l-4 BC v&i toa d6 dinh c(3; z;31 v) phuong trinh dulng cao AH'

r-2 u-:3 :-:l r-l Y'+ -:l phLrcrngtritrhduongphingiActrongBD )An lLrcvtl): rl, :=='::= d' l:5 - Lap

phrrong tr)nh ducrng thing chrla canh BC c0a f .1l]C vi tinh diQn tich crla A.4BC' '

Gqimp(P)qttaCvdvudrtgg6cvaiAH =+ P) -rlr'=>"P;: 't:i y -22+1:O

3:tP)a'{l ),+ Bll;4: 3) =phuangtrinh BC 'i':1.- )t:tt :}-2t"2:3

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

(Q) r -2u + z-2:0 =+ i{i2:2:1)=+ llt1"2'5) {Kldtrung didmcfiaCM)'

-.1 l: )'.-t - ,r,, :;l i, lTii : r.,,

l:i(' v0\ .l(l: ' l: l) vi hai dwong trung tuydn thn lucrt co

HT 160 'Long i<h6ng gtat.r r'6t h0 toa do

phil'on1_.q l'lnn la tu- !

I

Oxyz, chcr J-l

i, "I I

I

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

I

h rro'

' vIfT pHrroNG TRiNH rvrAr cAu

) Oy E/s: (:r: _ i,2

-iy *212 - (: - 3 j2 : l0 .

HT 162 Trong kh6ng gian vdi hO toa d6 Oxyz, cho hai dLrdng thing:

f,:zt [r: : - i

lA : t vA d, :

la : t Chr?ng minh r/r d, ch6o nhau ViSt phuong trinh mit cAu ISJ c6 duong kf n]r l) doan

[ : r l.: i,

vu6ng g6c chung cia dr rti

D /s: (t: - 2'i' - (.u - l')2 - i' :i - 2;! : 1.

Cdtt hot rlrarla tv:

a) d.,

.r -2

(dz)

2

, ,'2

DS: (,e) , (r -2):' +i, -il -,, - ,t,'

{ - ')

l

HT 163 Trong kh6ng gian vryi hA tQa d6 ax1,:r, cho hai dwcng thang: ,1, , ll i

- ,- l - ,i va

'Il

, -

=

: : Viet phutrng trinh nrat ciu co bin kinir :riro nhit tiSp xLic vcli ca hai rilrrrng trang rr va r.,.,. '.'.iat cau nhdn doa, vuang g6c chung cia hat dvitrct thang la clu,on,q kinh.

I

l-

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

[' :'

tt

- 1 -1 2

y-4 z-'2 ', .)

1

l::-2 - lJ*l _:

5 0

:l

l

i , .i - r a) ,i -l-_,-l-_),!t :t , i., ),1 : t

[r=* [e=0

HT164, Trongkhdnggian r,&i hO toa do Oxyz,clto d r'ong tlrang ij, i cri phuongtrinh {.2, :2i ,1/: t:z: L; (-rr; ia

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