Phương pháp giải toán hình học theo chuyên đề part 3

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

Phucmg phdp gidi Todii Hinh hoc theo chuyen de - Nguyen Phu Khdnh, Nguyen Tat Thu

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

2 2(2x-21)2 =19(x^+13) x<

2 o x =

-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 A i, A 2 , A 3 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; "

2) Do M€Ox=>M(m;0;0), N€Oy=i>N(0;n;0) F ' ' Suyra A M = ( 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^

n = •

n = •

22 + 3 N/23T

5 22-3^/23T

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:

2) Tinh chieu cao ve tu B ciia tam giac BCD va chieu cao ciia t i i 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

^ABCD 3) Ta CO M ^•1-1

2'2'

3 7]

~ ' ' 2 ' 2 j

- - - ^7 1 Suy ra CM= - ; - ; - 4

2

1 2^ 243 t +

2 0 f 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) V i A , B, C khong t h i n g hang nen A B C D la h i n h b i n h hanh k h i va chi khi

= V l - c o s B = - , 5 V 77

+) tanC,cosC cung dau tanC = |—I 1 =

k o s ^ C 38 4) Dien tich tam giac ABC la S = - B A B C s i n B = - 5 N / 7 7 - j — = VisT

2 s i n B 2 s i n B = 25

77

r = ^ = 2S

481 2V48T

p A B + BC + CA 15 + V77*

Chii y: Co the tinh dipn tich bang cong thuc S = A B , A C

5) Goi E, F Ian l u g t la giao diem ciia phan giac trong, phan giac ngoai goc A voi

d u o n g thSng BC

Theo h'nh cha't phan giac, ta c6 — = — =

^ ^ EC FC A C +) V i E nam trong doan BC nen EC = -2EB

G i a s u E{x^;y^;z^) thi EC(7 - X E ; 9 - y j : ; l - ),EB(5- X E ; 1 - y ^ ; - 2 - z ^ )

A ( 2 ; 3 ; l ) , B ( - l ; 2 ; 0 ) , C ( l ; l ; - 2 )

1) Tim toa do chan d u o n g 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 d u o n g tron ngoai tie'p cua tam giac ABC

4) Gpi G la trgng tam cua tam giac ABC C h u n g m i n h rang cac diem G, H , I nam tren mot d u o n g t h i n g

Phucntgphapgiai Todn Hinli iiQC theo chuyen de N^iiijcti I'hu Khanh, Nguyen Tat Thu

15 30 nen H G = 2 G I , tiic la ba diem G,

H , I nam tren mot duong thSng

Bai 6.1.3 C h o tam giac deu A B C c6 A ( 5 ; 3 ; - l ) , B ( 2 ; 3 ; - 4 ) v a diem C nam

trong m^t phang (Oxy) c6 tung do nho hon 3

1) T i m tpa dp diem D biet A B C D la tu di?n deu

2) T i m tpa dp diem S biet S A , S B , S C doi mpt vuong goc

Jiuong dan giai

Cty TNHl! MTV nVVH Kluuix Viet

V i C CO tung dp nho hon 3 nen C(l;2;0)

z = l - x

y - 1 6 - 5 x 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

x + y - 4 z = 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 € 0 x , D € 0 y , A ' G 0 z va AB = 1, A D = 2, A A ' = 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 M N 1 B D , M N 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

A N = AB + BN = AB + y.BD = ( l - y;2y;0) => N ( 1 - y;2y;0)

Theo gia thiet cua de bai, ta c6:

Vi M N la duong vuong goc chung ciia hai duong thang A'C,BD nen

Cty TNHH MTV DWH Khang Viet

d ( A ' C , BD) = M N = ^{1-x-yf + (2y-2xf + {3x - sf = 6 7 6 1

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; A D = 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 = - ^

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 ' ' ' '

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

Phuang phapgiai Toa,i Hinh hoc theo chuyen dj- NguySn Phii Khdnh, Nguyen Tat Thu

MN A MP =

• 4 ' 3 ' 2 nen M, N, P, Q khong dong phang"!

(MNAMP).MQ = — ^ 0

V M N P Q =^|(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 C C

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 ;

-C t y TNHH MTV DVVII Khang Vigt

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

2) Xac

4 + 2t;4 + t ; - - - 2 t

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 o t 2 + 4 t + — = 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 G D = V D A 2 - A G 2 = — = 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 P H l / O N G TRlNH M A T P H A N G

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 A 1 (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):

a b c

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 - Z o ) = 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

v a i A — - 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)//A (a)l(P) =>-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 : 4 x - y - 3 z + d=:0

^ M a t 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_

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) d i qua hai d i e m A ( ] ; 2 ; 1 ) , B ( - 2 ; 1 ; 3 ) sao cho khoang each t u C ( 2 ; - l ; i )

den (P) bang khoang each t u D (O; 3; 1) den (P)

L a i giai

1) Gia sir (P): ax + by + cz + d = 0 :h

T a c o A(2;-1;0),B(5;1;1) la d i e m chung eua (a) va (3)

V i (P) d i qua giao tuyen ciia hai mat p h l n g (a) va (|3) nen A , B e (P)

j ) Viet p h u o n g t r i n h mat p h c l n g (P) d i qua A va d j I

2) C h u n g m i n h rang d i va di cat nhau Viet p h u o n g t r i n h m a t phang (Q)

chua d i va d 2

JCffigidi-Taco: D u o n g t h ^ n g d i d i qua M ( l ; - l ; - 2 ) , VTCP u ^ = (2;1;-1)

D u 6 n g t h a n g d 2 d i q u a N ( 2 ; - 2 ; l ) , VTCP u ^ = ( l ; 2 ; - 4 ) 1) Taco: A M = ( 0 ; - 3 ; - 5 )

Do (P) d i qua A va d j nen n ^ = A M A u ^ = (-8;10;-6) Suy ra p h u o n g t r i n h (P): 4x - 5y + 3z - 3 = 0

2) Xet he p h u o n g t r i n h

l + 2t = 2 + t '

- l + t = - 2 + 2 t ' o

- 2 - t = l - 4 t ' Suy ra d i va d2 cat nhau tai E { 3 ; 0 ; - 3 )

^) Ta CO A 6 d j =^ A ( l + a; - 1 + 2a; 1 - a ) , B e da => B(-2b; - 1 - 4b; - 1 + 2b)

Suy ra A B = ( - a - 2 b - l ; - 2 ( a + 2b);a + 2 b - 2 ) , d a t x = a + 2b

S T U A B = >/13=>(X + 1)2+4X^ + ( X 2 ) ^ = 1 3 O X = 1 , X =

-Phuong phapgiai Todn Hinh hqc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu

o 3^/5 |b| = 2V5a2+12ab + 10b2 o IQa^ + 48ab - 5b^ = 0 <=>

Cty TNHH MTV DWH Khang Vi?t

Viet p h u o n g t r i n h mat phang (a) d i qua M , song song v o i A va (a) tao v o i

ba tia Ox, Oy, O z m o t t u dien c6 the tich bang 8

2) Viet p h u o n g trinh mat phang (P) d i qua M , v u o n g goc v o i ( P ) : x + y + z - 3 = 0

( a 2 ) : 2(5 - v/37)x + (7 + V37)y + 2(1 - V37)z - 24 = 0

Va (a3):2(5+>/37)x + ( 7 - V 3 7 ) y + 2 ( l + V37)z-24 = 0 '

2) V i mat phang (3) d i qua M nen p h u o n g t r i n h ciia (3) c6 dang: / '

* a ( x - l ) + b ( y - 2 ) + cz = 0 ^ a x + by + c z - a - 2 b = 0 (*) (^)'.; ' ^ voi a^ + b^+c^ > 0 •

I M a t khac, (P) 1 (P) nen rfp.rip = 0 o a + b + c = 0c=>c = - a - b

n,

|-2a + 2b + 3c|

Vr7.Va2+b2+c2 " 7 3 4 ( 7 7 ^ b 2 )

Phuontg phap giai Toan Hinh hoc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu

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

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 - 7 t Taco AB = (-7;2;8) nen phuong trinh AB:

Gpi K la hinh chieu cua E len AB, suy ra

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^ = U i , U 2 =(3;-l;-2) va di qua I nen jphuongtrinh : 3 x - y - 2 z + 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 ( d 2 ) 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 )

3' 3 '3)

Co duong thang BC la duong thang BA' di qua C va c6 VTCP

= CA' = i ( l l ; l ; 1 6 ) chon u ' = (ll;l;16)

3

Phuangphdpgiai Todn Hhih hoc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu

Chung minh rang khi m thay doi thi duang thang dm luon nam trong m o t

mat phang co'djnh Viet phuong trinh mat phang do

Cty TNHH MTV DWH Khang Viet

AB, AC, A D Ian lugt tai B',C',D' Viet phuang trinh mat phang (?) jji^'t t u dien A B ' 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 A B A A C = (-5; -5; 5) ( A B A A C ) A D = 20 ^ 0

Nen A , B, C, D la bo'n dinh cua t u dien

The tich tu dien A B C D la: V,XBCD = - ( A B A A C ) A D

Vi (a) song song vai A B , C D nen n„ =(3;9;-5) '

Vi MN, A B , C D Ian lugt n5m tren ba mat phang song song gom: Mat phang

di qua A B va song song vai C D , mat phang di qua C D va song song voi A B va mat phang (a) nen thco dinh li Talet dao trong khong gian ta suy ra:

t = 1, ta C O M(2; - 1 ; 0) = C nen ta loai truong hgp nay

t = - 1 , ta C O M(0; 3; 2) nen phuang trinh (a): 3x + 9y - 5z - 1 7 = 0

3) Ta C O G

2 2 Ggi A i la trong tam cua tam giac ABC, suy ra A j f_i.i.r

Phuang phdpgiai Todn Hinh hoc tlteo chuycn de - Nguyen Phu Khdnh, Nguyen TatThu

Ap dung bat dling thuc Co si ta c6:

[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 r i ^ = u j 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 = 0 0 BH.AC = 0

3a + 7b + c - l l = 0 2a-b + c + 5 = 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 - 7 z + 2 = 0

B a = 2b,tachon b = l=>a = 2.Phuangtrinh (P):2x + y - 3 z = 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)

Va A H < 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 ) : l l x - 4 y + 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

Phuang trinh (a): 4x -1 ly + 14z - 24 = 0

2) Vi (p) song song voi hai duong thing di, da nen ta c6

V^y phuang trinh (P): x - 4y + 3z + ^ = 0

Cty TNHH MTV D W H Khang Vift

3) Ta C O M(l - 2m; -1 + m; -2 + 2m), N(-l + n; n; -1 + n) Suy ra MN = (n + 2 m - 2 ; n - m + l;n-2m + l)

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

Phumig phdp gidi Todn Hinh hgc theo chuyen de- Nguyen Phil Khdnh, Nguyen Tat Tliu

x + 3z + l = 0 3x + 4 y - z + 5 = 0

» x + 2 y - 2 z + 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

Vay CO hai mat phang thoa man 2x + y - 2 z + 6 = 0, 2 x - 5 y - 1 4 z + 30 = 0

4) (a) song song voi mat ph3ng (Q): x - 2 y - 2 z - 3 = 0 nen c6 phuong trinh

(a): X ^ 2y - 2z + D = 0 Lay diem N(3;0;0) e (Q)

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

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 p h u a n g t r i n h mat phSng (P) tiep xuc v o i mat cau (S) va song song v6,

Jiuang dan gidi

M a t cau (S) c6 tam 1(2; - 3 ; 1), ban kinh R = 742

Vay p h u a n g t r i n h (P) la: x + 4y + 5z - 37 = 0 hoac x + 4y + 5z + 47 = 0

2) V i mat phang (a) chiia d j nen p h u o n g trinh (a) c6 dang

9 a - 2 b - 7 c 7a2+b2+c2 3

379,

• a = • -h, chon b = 670,3 = 379,c = 203

670 , Phuong t r i n h (p) la: 379x + 670y + 203z+ 1699 = 0

Bai 6,2.3 Trong k h o n g gian v o i h f true toa dp O x y z , cho 2 d i e m A(2;0;l), B(0; - 2 ; 3) va mat phSng (P): 2x - y - z + 4 = 0 T i m toa do d i e m M thuoc (P) saocho M A = M B = 3

(Trichcau6adethiDHKhdiA-20n) Jiie&ng dan gidi

'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])

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

M I vuong goc vai A va M I = 4\/l4

Jlitang ddn giai

Taco A cat (P) tai I(l;,l;l)

Diem M(x;y;3 - x - y ) e ( P ) =^ M I = ( l - x ; l - y;x + y - 2 )

Duong thing A c6 a = ( l ; - 2 ; - l ) laVTCP

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 + 2 t - 2 t - t - 2 = 0 < » t = - l = > C ( - l ; - l ; - l ) '

Diem M e A<»M(l + 2t;t;-2-t)=i>MC = >/6 o ( 2 t + 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 =

Phucnig phcip giai Todn Hiith hpc theo chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu

Bdi 11.2.3 Trong khong gian toa dp O x y z , 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

T a c o d(0;(R))-2<=> • = 2 <=> m = ±2N/2

Vl + 0 + 1

Vay ( R ) : x - z ± 2 V 2 = 0

Bdi 12.2.3 Trong khong gian voi h§ toa dp O x y z , cho tu dien A B C D c6 cac

dinh A(1;2;1), B(-2;1;3), C ( 2 ; - l ; l ) v a D ( 0 ; 3 ; 1 ) Viet p h u a n g 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:

T r u c m g h p ' p l : (P) di qua A , B song song voi C D

T a CO A B = (-3;-l;2), C D = (-2;4;0), suy ra n = r A B , C D ] = (-8;-4;-14) la

V T P T eua (P) Phuong trinh (P): 4x + 2y + 7z - 1 5 - 0

T r u a n g hgp 2: (P) di qua A , B v a cat C D tai I , suy ra I la trung diem cua

d:- •1 _ y _ z - 2 T i m toa dp hinh chieu vuong goc cua A len d va

• Gpi H ' la hinh chieu cua A len mp(P)

Khi do, ta eo: A H ' < A H => d(A, (P)) Ion nhat <=> H = H ' o (P) 1 A H

Suy ra A H = (1; 4; 1) la V T P T cua (P) v a (P) di qua H ^

-Vay phuong trinh (P): x - 4 y + z - 3 = 0

Bdi 14.2.3 Trong khong gian voi h ? tpa dp O x y z , 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 v a tim tpa dp true tam

t a m g i a c A B C 2) T i m tpa dp cua diem M thupc mat phang ( P ) : 2x + 2y + z - 3 = 0 sao cho

MA = M B = M C

JJu&ng ddn gidi

l ) T a c 6 : A B = ( 2 ; - 3 ; - l ) , A C = ( - 2 ; - l ; - l ) => r A B , A c ] = (2;4;-8) la mot VTPT

ciia m p ( A B C ) Phuong trinh m p ( A B C ) : x + 2 y - 4 z + 6 = 0

Gpi H(a;b;c) la tryc tam tam giac A B C

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

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:

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:

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

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

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 ,

»•.'

1 9 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)

M e (a) Diem M la true tam tarn giac ABC khi va chi khi AM.BC = 0

a^ •

1 9 4

•,C7t0 thoa m a n - + - + - = 1 (1)

a b c

Phuontg phap giai Todn Hinh hoc theo chuyen tie - Nguyen Phil Khdnh, Nguyen Tat Thu

Phuang trinh mat phang (a) can t i m la x + 9y + 4z - 98 = 0

Cdch 2;Goi H la hinh chieu ciia O tren mat phling ( a )

Vi mat phang (a) luon d i qua diem co'dinh M nen

d ( 0 , (a)) = O H < O M = ^98

Dau dang thuc xay ra khi H = M , khi do (a) la mat phang d i qua M va c6

vec to phap tuyen la O M ( l ; 9 ; 4 ) nen phuong trinh (a) la

a a a nen p h u o n g trinh (a) la: x + y + z - 1 4 = 0

1 9 4

• Truong hop 2: a = b = -c T u (1) suy ra - + = l<=>a = 6, nen phuong

a a a trinh (a) la x + y - z - 6 = 0

1 9 4

• T r u o n g hop 3: a = - b = c T u (1) suy ra + - = l o a = - 4 , nen phuong

a a a trinh (a) la x - y + z + 4 = 0

1 9 4

• Truong hg-p 4: a = - b = -c T u (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 - 1 4 = 0, va cac mat p h i n g

B Chung m i n h rang khi m thay doi, d u o n g thang d m luon nam trong mQt mat

phang (a) co dinh Viet phuong trinh mat phang do

Jiuang dan gidi

tD u o n g thang dm d i qua A ( 3 m + l ; l ; 6 m ) va c6 VTCP u = (2m; 1 - m ; l + 3m) Xet mat phang co d j n h ( a ) : ax + by + cx + d = 0 chua d m

K h i do: A e ( a ) l , V m

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

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 A B C D c6

A ( l ; 1 ; 1), B(2; 0; 2), C ( - l ; - 1 ; 0 ) , D ( 0 ; 3 ; 4 )

AB A C A D

Tren cac cgnh AB, A C , A D lay cac diem B', C', D ' thoa + + = 4

^ AB' A C AD- • 1) Viet p h u o n g trinh mat phSng (B'C'D') biet t u di^n AB'C'D' c6 the tich Ion nha't,

2 ) Viet p h u o n g trinh mat phJing (a) song song v o i AB, C D cat A C , BD Ian luot

Hfoe lap p h u o n g trinh d u o n g thSng A, ta can t i m m g t d i e m M ( x o ; y o ; z o ) ma

^ di qua va mQt VTCP u ^ ( a ; b ; c ) K h i do p h u o n g trinh cua A c6 dang: :

• Neu A la giao tuyen cua hai mat phang (a) va (P) t h i u = n^^ A np la m p t

VTCP cua A Trong do n ^ , n ^ Ian l u g t la VTPT ciia (a) va (P)

K h i chpn mat phang chua A, chiing ta can l u u y:

• N e u A d i qua M va v u o n g goc v o i d u o n g thang d t h i A n a m trong m^t phang (P) d i qua M va v u o n g goc v d i d

• Neu A d i qua M va cat d u o n g thMng d thi A nam trong mat phang (P) d i qua M va d

• Neu A d i qua M va song song v o i mat phang (a) t h i A n a m trong m^t phSng (P) d i qua M va song song v o i mat phang ( a )

_ 1

Vi du 1.3.3 Lap p h u o n g t r i n h d u o n g th5ng A biet: ^ ,

1) A d i q u a 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 ) : 2 x - y + 5 z - 4 = 0

^ A d i q u a M ( 1 ; 0 ; - 1 ) va vuong goc v o i hai d u o n g thang

Phicamg phdpgiai Todn Hiith UQC theo chuyen de - Nguyen Phii Khdtih, Nguyen Tat Thu

x = l - 2 t Phirong trinh tham so' cua A:<y = 2 - 2 t , t e M

z = l - t A j v

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 => M N = (-4; 7; 3) la mpt VTCP ciia A

x = - l - 4 t 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

Cdch 7;Gia sir u = (a;b;c) la mot VTCP ciia A

Vi A vuong goc voi di va d2 nen ta c6:

CtyTNHll MTV DVVH KIniiig Vict

2: Vi A vuong goc vol hai duong thang di, d2nen vec to

u = U j A U2 = ( - 6 ; - 3 ; - 2 ) la VTPT ciia duong thang A

x = l - 6 t

;;:tj;vlb'-z = - l - 2 t 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-,:,,

d i : y = t ; d j - y = 4 + 2 t ' ,

z = 4t z = l 2) A di qua M(-4;-5;3) va cat hai duong thang

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 = - 2 t , tG

z = t

2) Ta C O d i d i qua M i ( - l ; - 3 ; 2 ) va c6 u^ = ( 3 ; - 2 ; - l ) la VTCP Duong thang d2di qua M 2 ( 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) C O np = M M j A U j = (-4;0;-12) la VTPT

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 V T l ^ 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 , t e R 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 + 2 z - l l = 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 =

z = l + 2t 2x + y + 2 z - l l = 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 M H : Suy ra H(l + 2t;2 + t;l + 2t)

1 14 _8

9 ' 9 ' 9

_ x - 2 _ y - 1 _ z - 3 Phuong trinh I H :

255

Phuongphdpgiai Toan Hinh hpc theo chuyen de - Nguyen PhuTiluhih, Nguyen lat inW

Mat khac : cos 9 = 7 UA2 UA a - b + 2c|

3^2 3a + 4c = zVsa^ + 8 a c + 5c2 o 18(3a + 4cf = 49(53^ + 8ac + Sc^)

<=> 83a^ - 40ac - 43c^ = 0 <=> a = c,a = -—c

1) Viet phuong trinh duong thSng A cat hai duong thang di, d2 Ian lugt tai

M , N thoa M N = 3\f3 va M N song song voi mp (a): x - 2y + 3z = 0

2) Viet phuong trinh duong thang d di qua O va cat d j tai A sao cho mat

cau tarn A va di qua O tiep xiic voi mat phang (P): 5x + y - 4z -16 = 0

2) Taco A e d j 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))

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 A j Taco A n A j = A ( 2 - a ; - 1 + a; 2a), A n A j =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 A 1 A, ^ A 1 A, ^ nen < nen < A B l u ^ ' <=>< AB.u^^ = 0

AB.u^^

A l A2 A B l u , ^ AB.u^^ = 0 , do do

- l - 2 b - a + 3 - b - a - 2 - 2 b - 2 a = 0 J4a + 5b = 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 = A n A ^ =::>M(2-m;-l + m;2m)=>OM = ( 2 - m ; - l + m;2,m)

Vi A I A 2 nen MO.UA2 = 0<=> 2 ( 2 - m ) - ( - l + m ) - 2 m = 0<» m = 1

Suy ra U A = O M = (1;0;2)

x = t Phuong trinh A : < y = 0 , t e l R

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 d j 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

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;

d j 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

1 ^ ^„ 2>/4T ViT

S = -.IA.IB.sin(p = a = r:>a = l

2 42 42 aco: A e d j => A(3 +2t;3 + 2t;3 +1) => lA = (2t + 2;2t + 2;t +1)

Phuongphapgiai Toau Uhth hoc thco chuyen de- Nguyen Phi'i KhdnU, Nguyen Tai Thu

1 1 5 Phuong t r i n h A :

X — V —

z-3 _ 1 z-3 z-3

4 - 5 - 1 2) G o i d la d u a n g phan giac cua goc tao boi hai d u o n g thang d ^ d j

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 A B C la tarn giac vuong can tai A va S A l ( A B C ) H i n l i

chieu v u o n g goc cua A len SB, SC Ian lup-t la hai diem M , N va mat phang

( A M N ) C O p h u o n g trinh 2y + 3z + 3 = 0 Goi 13 la diem doi x u n g v o i A qu.i

trung diem E cua BC

1) Viet p h u o n g trinh d u o n g thang SD

V i D = SD n (ABC) nen tpa dp cua D la nghi^m ciia h^:

Cty TNHH MTV DWH Khang VilT

6 A va B, dong thai khoang each t u A den mat phang (a) bang S Viet

phuong trinh A , biet d i e m A c6 hoanh do duong

Vi du 9.3.3.Trong khong gian Oxyz, cho hai d u o n g thang

x = l - 2 t x = 3 - k

y = 2 + t va d2 :^ y = l + 2 k

z = 3 - 2 t z = 2 + 2 k

li