* Ta CO dj di qua M (O; 3; - l ) c6 VTCP u[ = (-1; 2; 3), d2 di qua N(4;0;3) coVTCP ^ = ( l ; l ; 2 ) , M N = (4;-3;4). N(4;0;3) coVTCP ^ = ( l ; l ; 2 ) , M N = (4;-3;4).
^ , U 2 M N = -23 7t 0 nen d ^ d j cheo nhaụ * Gpi A, B Ian lupt la giao diem cua A voi d p d j . * Gpi A, B Ian lupt la giao diem cua A voi d p d j .
A € d i = > A ( - t ; 3 + 2 t ; - l + 3t),Bed2=>B(4 + s;s;3 + 2s),t,sG] Suy ra AB = (s + t + 4;s-2t-3;2s-3t + 4) Ta CO A / /Ox nen s - 2 t - 3 = 0 2s-3t + 4 = 0' r s= - i 7 Phuong trinh A: t = -10 x = 10 + t y = -17 . z = -31 =^AB = (-23;0;0),Ă10;-17;-31). 071
Phuong phdp gidi Toan Hinh hoc thco chuyen de- Nguyen Phu Khdnh, Nguyen Tat Thu
x_-3 _ y + 2 _ z + 1 2 1 - 1 Bai 13.3.3. Trong he toa dp Oxyz cho duong thSng d :
mat phang (P): x + y + z + 2 = 0. Tim giao diem M ciia (d) va ( P ) . Viet phuon > trinh duong thang A nam trong (p) vuong goc voi (d) sao cho khoang each ti,
M den A bang 4 ^ .
Jiucomg đn gidi
Giao diem M ciia d,mp(P) la nghiem ciia he: x = l
y = -3 => M ( l ; - 3 ; 0 ) . z = 0
Goi (a) la mat phang chua d vuong goc voi mp(P). Theo gia thiét ta c6
A = ( P ) n ( Q ) , voi (Q)la mat phang vuong goc voi mp(a),mp(P) va khoang each t u M den (Q) bang V42 .
Ta CO Uj = ( 2 ; l ; - l ) ; n p = (1;1;1) Ian lugt la cac vec to chi phuong, phap tuyen cua d,mp(P).
Vec to phap tuyen ciia mp(a): = [ u ^ ; iC] = (2; -3; 1). x - 3 _ y + 2 _ z + l
2 ~ 1 " - ] < : : > x + y + z + 2 = 0
Vec to phap tuyen cua mp(Q):nq = [ n „ ; n p ] = (4;l;-5) Phuong trinh mp(Q) :4x + y - 5 z + a = 0
4.1-1.3 +a ^ 4 ^ + 1 + 5^ 1 + a - 4 2 o a = 4 1 ; a = - 4 3 . Theo gia thiét ta c6: d(M,(Q)) = Vii o = 742
* Voi a = 41=>(Q):4x + y - 5 z + 41=:0
Phuong trinh duong thMng A :
x = -13 + 2t y = l l - 3 t z = t * Voi a = -43=>(Q):4x + y - 5 z - 4 3 = 0
x = 15 + 2t Phuong trinh duong thang A : y = -17 - 3t,
z = t
Bai 14,3.3. Trong khong gian Oxyz cho hai duong thang
^ 1 - 1 1 2 3 _ ^ _ ^ •
Cty TNHH MTV DWH Khang Vỉt
iChung minh rang hai duong thSng Aj va A2 cheo nhaụ Viet phuong trinh ^y,^ng vuong goc chung cua hai duong thang dọ
2) Cho A, B la hai diem di dong tren Ai thoa AB = 4. Viet phuong trinh duong j^ang di qua O va cat A2 tai C sao cho tarn giac ABC c6 dien tich nho nhat. 5) Co ton tai hay khong mot t u dien deu c6 hai dinh thuoc Ai va hai dinh thuoc
>^2. Néu CO hay tim toa do cac dinh dọ
4) Viet phuong trinh duong thSng d di qua P(l;4;5) va cat Aj^, Ag Ian luot tai j ^ , H sao cho I ' H = 2PK.