M: M2 =(1; 1; 2) nen hai duong thang do song song.
12 2' a) Tim toa dp giao diem ciia d vdi (a).
b) Tinh cosin ciia goc hpp bdi d va (a).
c) Viet phucmg trinh mat ciu co tam thuoc d, tidp xiic vdi hai mat phing (a) va (Oxy).
Giai
a) Gpi M la giao didm ciia d vdi (a). Toa do ciia M la nghiem ciia he '2x - y + 2z + 1 = 0
phucmg trinh: x - l _ y - 1
•=>M - ; 2; - 1 ,2
1 2 - 2
b) Vecto phap tuydn ciia (a) la n = (2; - 1 ; 2), vecto chi phucmg ciia d la
u = (1; 2; -2). Gpi cp la goc giua d va (a) thi:
n.u
sin cp = ~
n . u
| 2 - 2 - 4 | 3.3 Ta co sin2cp + cos2cp = 1 coscp
Do (S) tiep xuc voi (a) va mat phang (Oxy) nen:
An < An n ^^ |2(1 +1) - (1 + 2t) - 4t + l |
d(I, (a)) = d(I, Oxy)) <=> l- 1 2t
<=> |2t -1\ = |3t| <r> t = - 1 hoac t = -
Voi t = -1 thi (S) co tam 1(0; -1; 2) va ban kinh R = 2 nen (S) co phuong trinh x2 + (y + l )2 + (z - 2)2 = 4.
Voi t = - thi (S) co tam I
5 v ' 6 x 5 6 7 5;5 r 2 5
- ; - — va ban kinh R = - nen (S) co
z + • 25 4_ phuong trinh
Vi du 17: Trong khong gian toa do Oxyz cho duong thang d:
: 1+ 2t : -1 + t 2-t Goi d' la giao tuyen cua hai mat phang:
(a): 3y - z - 7 = 0 va (a'): 3x + 3y - 2z - 17 = 0. a) Chung minh d, d' cheo nhau va vuong goc voi nhau.
b) Viet phuong trinh mat phang (P) di qua d' va vuong goc voi d. Tim toa do giao diem H cua d va (P).
c) Mot mat phang (Q) thay doi, luon song song voi mp(Oxy). cat d, d' lan luot tai M , M'. Tim quy tich trung diem I cua doan M M ' .
Giai
a) Duong thang d' la giao tuyen cua hai mat phang co vecto phap tuyen la n = (0; 3; -1) va n ' = (3; 3; -2) nen d' co mot vecto chi phuong la: u7 = [ n , n ' ] = ( - 3 ; - 3 ; - 9 ) hay (1; 1; 3)
Vecto chi phuong ud ciia d la ud = (2; 1; -1). Ta co ud ud. = 0 nen d i d ' . f 3 ( - l + t) - (2 - t) - 7 = 0
He
0 vo nghiem nen d va d' khong
3(1 + 2t) + 3 ( - l + t) - 2(2 - t) - 17 co diem chung. Vay chiing cheo nhau.
b) Cho y = 0 thi z = - 7 , x = 1, ta co A ( l ; 0; -7) e d'. V i d 1 d' nen mat phang di qua A va vuong goc voi d se di qua d'.
Vay phuong trinh mat phang (P) la:
2(x - 1) + (y - 0) - (z + 7) = 0 <=> 2x + y - z - 9 = 0. Toa do giao diem H(x; y; z) cua d va (P) thoa man he
x = 1 + 2t H 13 2 1 3 1 3! 3 y = - 1 + t 5 J => t = - z = 2 - t 3 2x + y - z - 9 = 0
c) Mat phang (Q) song song voi mp(Oxy) nen co phuong trinh: z = m (m * 0). Toa do giao diem M(x; y; z) ciia d va (Q) thoa man he:
x = 1 + 2t
y ~ 1 + t => M(5 - 2m; 1 - m; m) z = 2 - t
z = m
Toa do giao diem M'(x; y; z) cua d' va (Q) thoa man he: ' 3 y- z- 7 = 0 , . ( 1 0 + m 7 + m 3x + 3 y - 2 z- 1 7 = 0 => M ; ; m J L 3 3 z = m ^ T I. • Tf 25 - 5m 5- m
Goi I la trung diem cua M M thi I ;— -—; m
V, 6 3
Vay quy tich cua I la duong thang co phuong trinh tham so _ 25 5m
X ~~6~ ~6~
- - — bo di diem A( —; - ; 0 | (ung voi m = 0). y =
3 3 {6 3
z = m
V i du 18: Trong khong gian Oxyz cho diem S(2; 0; -1) va vecto u = (1; 0; 1). Goi A la duong thang qua S co vecto chi phuong u .
a) Chung minh rang tap hop cac diem M thuoc mat phang Oxy ma goc giira A va ducmg thang SM bang 60° la mot hypebol (H). Hay tim toa do cac tieu diem ciia (H).
b) Goi (Pi) va (P2) la cac mat phang di qua S va chiia mot trong hai duong tiem can ciia (H). Chimg minh rang tich cac khoang each tir mot diem thuoc (H) den hai mat phang (Pi) va (P2) la mot dai luong khong doi.
Giai
a) M £ mp(Oxy) => M(x; y; 0), SM = ( x - 2; y ; l ) Goc giira ducmg thang SM va A bang 60°
1 .AO I , 5 w | K x - 2 ) + 0.y + 1.1I x2 y2 ,
<=> - = cos60 = cos(SM, u) = , 1 _ 1 o — = 1. 2 1 1 ^/^2y^2 V(x - 2)2 + y2 + 12 3 3 Vay trong mat phang toa do (Oxy), tap hop cac diem M ma goc giua ducmg thang SM va A bang 60° la mot hypebol (H) co phuong trinh la: (H): 2 L _ y _ = i , z = o.
3 3
Cac tieu diem: F i ( S ; 0; 0), F2( V6 ; 0; 0)
b) Hypebol (H) co hai duong tiem can Ai va A2 lan luot co phuong trinh: y = ±x, z = 0.
Phucmg trinh ciia mat phang (P2) la: x + y + 2z = 0.
Gia sir N la mot diem bat ki thuoc (H) thi N co toa do (X; Y; 0) va 3 3 I X-Y + 2.O| |X + Y + 2.O| Taco: d(N, (P,)).d(N, (P2)) 1 <=> X2 - Y2 = 3 v l1 + r + 2" 4 r + r X2 - Y khong doi. 6 6 2
Vi du 19: Trong khong gian Oxyz cho hai diem A ( l ; 0; 0) va A'(—1; 0; 0). Goi A la duong thang qua A va song song voi Oz, A' la duong thang di. qua A' va song song voi Oy.
a) Tim tap hop cac diem M nam trong mat phang (Oxy) va each deu A va A'. b) Tim tap hop cac diem M nam trong mat phang (Oyz) va each deu A va A'.
Giai
a) True Oy co VTCP J = (0; 1; 0), true Oz co VTCP k = (0; 0; 1).
Gia sit M la mot diem thuoc mat phang (Oxy) thi M(x; y; 0). Ta co: d(M, A) = d(M, A') » [AM,k] [ A ' M , j]
J o y2 + (x - l )2 = (x + l )2 <=> y2 = 4x.
Vay tap hop cac diem M nam trong mat phang (Oxy) each deu A va A' la mot parabol co phuong trinh: y2 = 4x, z = 0.
b) Diem M e mp(Oyz) thi M(0; y; z). Ta co:
d(M, A) = d(M, A') ^ [ A M , k ] | | [ A ' M , j ]
<=> y2 + 1 = z2 + 1 <=> y = ±z.
Vay tap hop cac diem M e mp(Oyz) each deu A va A' la hai ducmg thang
fx = 0
lan luot co phuong trinh: d: x = 0 y = t d' z = t
y = t z = - t
Vi du 20: Trong khong gian Oxyz cho tap hop cac mat phang ( am) co
phuong trinh la mx - 2(m - l)y + (m + l)z - 1 = 0.
a) Chung to rang cac mat phang (ctm) di qua mot duong thang co dinh A.
x = l - 2 t
b) Cho duong thang d voi phuong trinh tham so y = 3t Chimg to
z = -2 - t
c) Lap phucmg trinh 2 mat phang lan luot chua mot duong thang d hoac A va chua ducmg \-uong goc chung ciia chung.
Giai
Phuong trinh cac mat phang ( am) co the viet thanh: 2y + z - 1 + m(x - 2y + z) = 0
f 2y + z - 1 = 0 Dang thuc nay diing vai moi m nen ta suy ra: - 2y + z = 0
He phuong trinh nay xac dinh mot duong thang A co dinh la giao tuyen
cua 2 mat phang 2y + z - 1 = 0, x - 2y + z = 0.
A co VTCP v = [ n 7 m j = (4; 1;-2) va di qua B( - l ; 0; 1).
, , x +1 y z — 1
Vay cac mat phang ( am) di qua duong thang co dinh A: —— = - = ——
b) d qua A(l; 0; -2) va co VTCP u = (-2; 3; -1)
Ta co [ u , v ] . AB * 0 nen d va A cheo nhau.
c) Ducmg vuong goc chung IJ co VTCP: a = [ u , v ] = (-5; - 8 ; -14)
Mat phang (P) chiia d va IJ nen co VTPT
mj = [ u , a ] = (-50; -23; 31) va di qua A ( l ; 0; -2) nen co phucmg trinh: -50(x - 1) - 23(y - 0) + 31(z + 2) = 0 hay 50x + 23y - 31z - 112 = 0. Mat phang (Q) chiia A va IJ nen co VTPT
n ^ = [ v. a ] = (-30; 66; -27) va di qua B ( - l ; 0; 1) nen co phucmg trinh: -10(x + 1) + 22(y - 0) - 9(z - 1) = 0 hay lOx - 22y + 9z + 1 = 0. Chii y: Duong vuong goc chung IJ la giao tuyen cua 2 mat phang (P), (Q).
V i du 21: Trong khong gian Oxyz, xet mat phang
(am): 3mx + 5 V l-m2 y + 4mz + 20 = 0, m e [ - 1 ; 1]
a) Tinh khoang each tir goc O toi mat phang ( am) .
b) Chiing minh rang voi moi m e [ - 1 ; 1], ( am) tiep xiic vai mot mat cau
co dinh.
c) V o i gia tri nao ciia m, hai mat phang ( am) va (Oxz) cat nhau? Khi m
thay doi, chiing minh rang cac giao tuyen do song song. Giai
a) d(0; ( am) ) = 2 0 = ™L = 4
V 9 m2+ 2 5 ( 1 - m2) +16m2 v25
b) Tir cau a) suy ra rang: Khi m thay doi, cac mat phang ( am) luon tiep xiic
vai mat cau co dinh tam O va ban kinh bang 4.
c) Mat phang ( am) co vecto phap tuyen n = (3m; 5 v'l-m2; 4m) vi vay ( am)
cat mp(Oxz) khi va chi khi m * 0. Khi do giao tuyen Am cua mp(am) va
mp(Oxz) la giao tuyen ciia hai mat phang:
3mx + 5 V l - m2
y + 4mz + 20 = 0 va y = 0.
V i m * 0 nen u = (4; 0; -3) la mot vecta chi phuong cua Am.
Do u khong phu thuoc vao m nen cac giao Uiyen Am song song voi nhau khi m thay doi.
Vidu 22: Cho tu dien ABCD co 4 dinh xac dinh: OA = - 2 j OB = 7 3 . F + J
O C = - 7 3. r+ J CTD = 2 7 2 . k .
a) Chung minh ABCD la tii dien deu.
b) Viet phucmg trinh tham so cua dudng vuong goc chung A ciia hai dudng thang A B va CD.
c) Tinh goc giua dudng thang A va mat phang (ABD). Giai
Taco A(0; -2; 0), B(73 ; 1; 0), C ( - 7 3 ; 1; 0), D(0; 0; 2 7 2 )
a) A B2 = 3 + 9 = 12, AC2 = 3 + 9 = 12, A D2 = 4 + 8 = 12.
Tuong tu BC2 = CD2 = DC2 = 12.
Do do 6 canh ciing bang 2 73 nen ABCD la tir dien deu.
b) V i ABCD la t i i dien deu nen dudng vuong goc chung A ciia AB va CD la
doan thang noi cac trung diem I va J tuong img ciia A B va CD. Ta co: I 73" - i ; 0
2
73 . 1
2 '2 72
Vecto chi phuong ciia A la u = (-73 ; 1; 72 ) 73" Vay phuong trinh tham so ciia A la:
— - V 3 t y
z = 72t
i + t 2
c) Mat phang (ABD) co vecto phap tuyen:
n = [ A B , AT3] = (672 ;-276 ;273) hay ( 3 V 2 ; - 7 6 , 73). Vay goc nhon cp giira A va mat phang (ABD) xac dinh bdi
376 73 3 3
sin cp = cos(u, n) = cp a 36°16'
373.76
Vi du 23: Trong khong gian vdi he toa do Oxyz cho bon diem: A(6; - 2 ; 3), B(0; 1; 6), C(2; 0; -1), D(4; 1; 0).
a) Chung minh rang A, B, C, D la bon dinh ciia mpt tu dien. b) Tinh the tich tii dien ABCD.
c) Viet phuong trinh mat cau ngoai tiep tu dien ABCD. Xac dinh toa dp tam I va ban kinh ciia mat cau.
d) Viet phucmg trinh dudng tron qua ba diem A, B, C. Hay tim toa dp tam va ban kinh ciia no.
Giai
a) AB = (-6; 3; 3), AC = (-4; 2; -4), AD = (-2jjJ; -3) => [ A B , AC] = (-18; -36; 0) => [ A B , A C ] . AD = -72 * 0 Vay A, B, C, D khong ddng phang nen la 4 dinh ciia tii dien.
b) V ABCD - | [ A B , A C ] .AD | = J - ^ l = 12.
c) Goi (S) la mat cau ngoai tiep t i i dien ABCD. Phuong trinh ciia (S) co dang: x2 + y2 + z2 + 2ax + 2by + 2cx + d = 0.
12a - 4b + 6c + d = -49 2b + 12c + d = -37 4a - 2c + d = -5 8a + 2b + d = -17
(S) qua A, B, C, D nen ta co he: <=> i
a = -2 b = 1 c = -3 d = -3
Vay phuong trinh cua mat cau (S) la:
x2 + y2 + z2 - 4x + 2y - 6z - 3 = 0 co tam 1(2; - 1; 3),
R = V22 + l2 + 32 + 3 = Jvf
d) Mat phang (ABC) co phap tuyen
n = [ A B , A C ] = ( - 1 8 ; - 3 6 ; 0 ) h a y ( l; 2; 0 ) va qua A nen (ABC):
x + 2y - 2 = 0.
Phuong trinh duong tron qua ba diem A, B, C la:
x2 + y2 +z2 - 4x + 2y - 6z - 3 = 0
[x + 2y - 2 = 0
Tam ciia mat cau (S) co toa do 1(2; - 1; 3). Duong thang A qua I va vuong
x = 2 + t
goc voi mp(ABC) co phuong trinh: < y = - 1 + 2t.
Iz = 3
Tham so t img voi giao diem H ciia A va mp(ABC) la tam duong tron giao tuyen: (C): (2 + t) + 2 ( - l + 2t) - 2 = 0 => t H 12 1 5 ; 5 /405 ;3 . Duong tron (C) co ban kinh r = V R2 - I H2
o
V i du 24: Trong khong gian Oxyz, cho hinh lang tru diing A B C . A | B , C i voi A(0; -3; 0), B(4; 0; 0), C(0; 3; 0), Bi(4; 0; 4)
a) Tim toa do dinh A , , Ci. Viet phucmg trinh mat cau tam A va tiep xiic voi mp(BCCiBi).
b) Goi M la trung diem ciia A i B | . Viet phuong trinh mat phang (P) di qua A , M va song song voi B G. Mat phang (P) cat duong thang A i d tai N . Tinh do dai M N .
Giai
a) Ta co: AA^ = BB^ = CC^ = (0; 0; 4) nen A,(0; - 3 ; 4), C,(0; 3; 4) va BC = ( - 4 ; 3 ; 0 ) n e n [ B C , BB-J = (12; 16; 0) la VTPT ciia (BCCiBi) (BCCiBi): 12(x - 4) + 16y = 0 <=> 3x + 4y - 12 = 0
24
Ban kinh R = d(A; (BCCiBi)) = — nen phuong trinh mat cau: 5 2 / „,5 2 576 x2 + (y + 3)2 + z2 = — b) Ta co: M(2; -|; 4), AM = (2; |; 4), BC\ = (-4; 3; 4) (P) co VTPT n = [AM, BC^ ] = (-6;-24; 12) Suy ra (P): x + 4y - 2z + 12 = 0 fx = 0 AjC, = (0; 6; 0) nen A1C1 :j y = -3 + t |z = 4
N thuoc A,C| nen N(0; -3 + t; 4) va N thuoc (P) nen t = 2, do do N(0; - 1 ; 4). Vay MN = —
•J 2
Vi du 25: Trong khong gian toa do Oxyz cho mat phang
(a): Ax + By + Cz + D = 0, ABC * 0 va diem M0( x0; y0; z0) khong thuoc (a). Cac duong thang qua M0 lan luot song song voi cac true toa do Ox, Oy, Oz c i t (a) tai M i , M2, M3.
a) Xac dinh diem M i va tinh doan M0M i b) Tinh thd tich khdi tii dien M0M i M2M3.
Giai
a) Goi di la duong thing qua M0( x0; y0; z0) va song song voi true Ox thi di co vecto chi phuong la i = (1; 0; 0).
x = x„ + t
Ta co phuong trinh ciia di la: d]: y = y
0
z = z„
Goi M i la giao diem ciia di voi mp(a). Toa do (x; y; z) ciia Mi thoa man he: x = x„ +1 y = y0 z = z0 Ax + By + Cz + D = 0 Axn + Byn + Czn + D 'MJ Xp °- ^ 2 ; yo;X o
b) Goi d2 la ctuong thang di qua M0 va song song vdi Oy. d2 cat (a) tai M2 thi: M0M2 = A x0 +B y0 +C z „+D