M: M2 =(1; 1; 2) nen hai duong thang do song song.
b)Diem M(x;y;z) thuoc Am khi toa do cii aM la nghiem ciia he: fmx + y mz 1 =
fmx + y - mz - 1 = 0
[x - my + z - m = 0
Khtr z tir he phuong trinh, ta duoc phuong trinh: 2mx + (1 - m2)y - 1 - m2 = 0.
Day la phuong trinh ciia mat phang ( am) chiia Am va song song vdi true Oz. Do do, khoang each giua Am va Oz bang khoang each tu goc O(0; 0; 0) thuoc Oz tdi mp(am).
Vay khoang each do bang:
-2| 9 1 , ™2 1: khong doi. m m V ^ m2 + ( l - m2)2 V m5 2m' 1
c) Toa do giao diem M ciia Am va mp(Oxy) la nghiem ciia he:
mx + y = 1 mx + y- mz - 1 = 0 x - my + z - m = 0 <=> z = 0 my 0 m
Binh phuong hai ve ciia hai phirong Uinh dau ciia he roi cong lai, ta duoc: x2 + y2= l
z = 0
V§iy tap hpp cac diem M la dudng tron tam O, ban kinh bang 1 trong mat phang toa dp (Oxy).
Vi du 18:
a) Tim tap hop cac diem M each deu ba d i i m A ( l ; 1; 1), B ( - l ; 2; 0), C ( 2 ; - 3 ; 2 ) .
b) Tim tap hop cac d i i m M each deu hai true toa do Ox, Oy va diem A ( l ; 1;0).
Giai
a) Biem M(x; y; z) each d i u ba diim A, B, C khi va chi khi:
M A = M B = MC [MA = MB [ A M 2 = B M2 | M A = MC A M2 = C M2 2 • z2 <=> | ( x - l ) 2 + ( y - l )2 + ( z - l )2 =(x + l )2 + ( y - 2 ) [ ( x - l )2 + ( y - l )2 + ( z - l )2 = ( x - 2 )2 +(y + 3)2 + ( z - 2 )2 f-4x + 2y - 2z - 2 = 0 [2x - 8y + 2z - 14 = 0 <=> 2 x - y + z + l = 0 x - 4 y + z - 7 = 0
Tap hop diem M(x; y; z) la duong thang giao tuyen cua hai mat phang tren. Bat y = t thi x = -8 - 3t va z = 15 + 7t.
x = -8 - 3t Vay tap hop la true cua tam giac ABC co phuong trinh la: I y = t
z = 15 + 7t b) Xet diem M(x; y; z). Khi do khoang each tu M toi true Ox va Oy la:
[ O M j ] _ | [ Q M , r ] Vy2 + z2; d = Vx2 + z2 j <=> < f 2 2 x = y [x2 - 2(x + y) + 2 = 0 Va M A = V(x -1 )2 + (y -1 )2 + z2 Ta co dx = dy = M A ^ [ y2 + z2= x2+ z2 [y2 + z2 = x2 + y2 + z2 - 2(x + y) + 2 K h i x = y, x2- 4 x + 2 = 0 ^ > x = 2 ± ^ .
Trong truong hop nay, tap hop cac diem M la nhung diem (x; y; z) thuoc
x = 2 + s/2 [x = 2 - V2
y = 2 + V2 | y = 2 - V2 z = t
2 duong thang:
Khi x = - y , x2 + 2 = 0. Dieu nay khong xay ra.
Vi du 19: Cho duong thang di di qua diem Mt( 0 ; 0; 1) song song voi Oy va duong thang d2 di qua diem M2( 0 ; 0; -1) song song voi Ox.
Tim tap hop cac diem M nam trong moi mat phang toa do va each d i u d,,d2.
Giai M(x; y; z) each deu di va d2 khi va chi khi:
d(M. di) = d(M. d2) o V ( z - D2 + x2 = f z + If + y2 <=> x2 - 2z = y2 + 2z o x2 - y2 = 4z.
Xet M E mp(Oxy). khi do z = 0 suy ra x2 - y2 = 0. (adsbygoogle = window.adsbygoogle || []).push({});
Vay quy tich diem M la cap duong thang y = ±x nam Uong mat phang z = 0. XeT M e mp(Ovz). khi do la x = 0. Quy tich d i i m M la ducmg parabol y = -4z nam trong mat phang x = 0.
Xet M E mp(Oxz), khi do y = 0. Quy tich d i i m M la duong parabol x - 4z nam trong mat phang y = 0.
V i d u 2 0 : c h o hai d i i m A(3; ljJO; B(7; 3; 9) va mp(a): x + y + z + 3 = 0. Tim diem M tren (a) d l | MA + MB | dat gia tri nho nhat.
Giai
Goi I la trung diem ciia doan A B 1(5; 2; 5)
T a c 6 M A + M B = 2 M I = > | MA + MB | = 2MI.
Vay | MA + MB | nho nhat c=> M I nho nhat o M l a hinh chiiu vuong goc ciia I tren mp(cx).
Toa dp ciia M(x; y; z) la nghiem ciia he: x = 5 + t
y = 2 + t
<z = 5 + t ^ t = - 5= > M ( 0; - 3; 0 ) .
x+y+z+3=0
V f q u 2 1 : Cho hai d i i m A(3; 1; 0), B(-9; 4; 9) va mp(a): 2x - y + z + 1 = 0. Tim toa dp diem M tren (a) sao cho | M A - M B | dat gia tri Ion nhat.
Giai
D | t f(x; y; z) = 2x - y + z + 1 thi fTxA; yA; zA).f(xB; yB; zB) < 0 nen hai diem A. B o khac phia doi voi mat phing (a).
Gpi A' la diem doi xung ciia d i i m A qua mat phing (a). Taco: I M A - M B I = | M A ' - M B | < A'B (Khong d6i).
A'H: x = 3 + 2t, y = 1 -1 , z = t nen H(3 + 2t;
= > H ( 1 ; 2 ; - 1 ) . Do do A ' ( - l ; 3 : - 2 ) .
x = - 1 + 8t
Duong thang A'B co phuong trinh
t; t) thuoc (a) suy ra t = -1
Toa dp diem M(x; y; z) thoa man he: x = - 1 + 8t
y = 3 - t z = -2 - l i t 2 x - y + z + l = 0
v' d¥ 22: Cho hai diem A ( l ; 2; -1), B(7; - 2 ; 3) va dudng thing d co phuong Uinh: *±1 = I z l = I z l
3 -2 2
a) Chung to rang duong thang d va duong thing AB cung thuoc mot mat phang.
b) Tim diem I thuoc d sao cho IA + IB nho nhit. Giai
a) d co VTCP u = (3; - 2 ; 2) va di qua M ( - l ; 2; 2). Ta co AB = (6; - 4 ; 4) = 2u va M khong thuoc d nen duong thang A B song song voi d.
b) Phuong trinh mat phing (P) di qua A va vuong goc voi d: (P): 3 ( x - 1) - 2(y - 2) + 2(z + 1) = 0
<=> 3x - 2y + 2z + 3 = 0.
Biem H thuoc d nen H ( - l + 3t; 2 - 2t; 2 + 2t) va thuoc (P) nen t = 0 = > H ( - l ; 2 ; 2 ) . (adsbygoogle = window.adsbygoogle || []).push({});
Do do diem ddi xung A qua d la A'(-3; 2; 5). Ta co I A + IB = IA' + IB > A'B: khong ddi.
Do do I A + IB be nhit khi I = A'B n d ma A B song song d nen I chinh la trung diem A'B. Vay 1(2; 0; 4).
fx = 1 + 2t Vi du 23: Cho A(2; - 2 ; 1), B(0; 2; -3). Tim didm M thuoc d: y = 2 - t sao
z = 1 + t
cho M A + M B be nhit.
Giai Ta tim hinh chidu A', B' cua A , B len d.
Ta co M bit ky thuoc d thi M ( l + 2t; 2 - t; 1 + \
A M2 = (2t - l )2 + (4 - t )2 + t2
= 6t2
- 12t + 17
= 6(t - l )2 + 11 > 11 A M be nhit khi t = 1, khi do M la hinh chidu A'(3; 1; 2)
Tuong tu B M2
= 6t2
+ 12t + 17 = 6(t + l )2
+ 11 > 11. B M be nhit khi t = 1, khi do M la hinh chidu B ' ( - l ; 3; 0).
Tren mp(A, d) lay didm B, sao cho B, va A khac phia ddi vdi d B , B ' l d B,B' = BB'.
Vdi moi M thuoc d: MA + MB = MA + MBX > A Bi : khong ddi, do do M A + M B be nhat khi M la giao didm ciia ABi vdi d .
Ta co A A ' // B,B' nen M chia doan A'B' theo ti: k = - AA
B j B '
hi
Jii = - l = ^ M ( l ; 2 ; 1).
V i du 24: Cho ba diem A(2; 0; 1), B(2; - 1 ; 0), C ( l ; 0; 1). Tim tren duong x = t
thing (d): < y = 2t diem S sao cho ISA + SB + SC| dat gia tri nho nhat. z = 3t
Giai