Neu c - 0 taco d(0,(P)) =
7 ^
Neu c . 0 , ta co: d(0,(P)) = 4±ib£±i£l = JiiUlii, t = ^ V5b2+8bc + 5c2 VSt^+St + S c
Y ' . . < t^ + 4 t + 4
Xet ham so f(t) = — , ta co: 5t^ + 8 t + 5
(St^ + 8t + 5f 2
Do f(-2) = 0, f
V 2 ,
1
= — l i m f(t) = - nen taco maxf(t) = f
JO t->±:o 5 I 2
Suy ra d(0,(P)) ion nhat khi va chi khi - = - 1
c 2 298
Cty INllH MIV UVVtl i^^nang vie
Ta chon b = 1, c = - 2 a = 2 .
Vay phuong trinh (P): 2x + y - 2z - 3 = 0 .
Cach 2: Giai bang p h u o n g phap hinh hQC
Goi K la hinh chieu ciia O len A, suy ra
I K ( l + t ; l + 2t;2t), O K = ( l + t ; l + 2t;2t) I V i O K I A nen O K . u ^ = 0 1 + 1 + 2(1 + 2t) + 2.2t = 0 <^ t = - | I V i O K I A nen O K . u ^ = 0 1 + 1 + 2(1 + 2t) + 2.2t = 0 <^ t = - | Suy ra K 2 1 _ 2 ^ 3 ' 3 ' 3 , OK = '2 1 2^ .3 3 ' 3 ,
Goi H la hinh chieu ciia O len (P), ta co: d(0,(P)) = O H < O K = 1 Dang thuc xay ra khi H = K .
Do do (P) each O mot khoang Ion nhat khi va chi k h i (P) d i qua K va vuong ' 6c v o i OK. Tir do ta suy ra phuong trinh cua (P) la: 2x + y - 2z - 3 = 0 .
) Cach 1: Giai bang phuong phap dai so
V i (Q) chua A nen phuong trinh ciia (Q) co dang:
ăx - 1 ) + b(y - 1 ) + cz = 0 v o i â + b^ + c^ > 0 va a + 2b + 2c = 0 \i cp la goc giua hai mat phang (Q) va ( a ) , ta co:
cos(p = a - 2 b + 2c
. .
"Q " a sVâ+b^+c^ 3\/5b2+8bc + 5c2
N§'u c = 0, ta CO cos(p = 0
4 1
Neu c;tO taco cos(p = - . ,
3 7 5 t ^ + 8 t + 5
, v o i t = - . b
Ta CO 9 nho nhat <=> cos 9 Ion nhat o 5t^ + 8t + 5 nho nhat.
Ma 5 t 2 + 8 t + 5 = 5 ^ 4^^ 9 9 • 4
+ - > - . Dang thuc co khi t = — .
5 5 ^ 5
t + '
c 4
Do do, 9 nho nhat k h i - = — t a chon c = 4, b = - 5 , a = 2 b 5
Vay p h u o n g trinh ( Q ) : 2x - 5y + 4z + 3 = 0 . Cach 2: Giai bang p h u o n g phap hinh h<?c.
Goi d la d u o n g thang di qua A va vuong goc voi (a) x = l + t
Ta CO p h u o n g trinh d : y = l - 2 t , l a y C ( 2 ; - l ; 2 ) e d , C ^ A
uung pnupgllll 1 uun nimi nor lneo cnuyen ae - Nguyen fnu Khanh, Nguyen Tat Thu
Goi H , K Ian lug-t la hinh chieu cua C len (Q) va A , k h i do cp = B C H va
. A H A K - sin cp = sin A C H = > . A C A C
A K
M a —— khong doi, nen suy ra
ẠC
(p nho nhat <=> H = K hay (a) la mat phang d i qua A va vuong goc voi mat phang ( A C K ) .
Mat phang ( A C K ) d i qua A va vuong goc voi (a) nen ní = [ n ^ , ^ ] = (-8;0;4) l a V T P T c u a ( A C K ) .
Do (Q) d i qua A va vuong goc voi (ACK) nen = [ i h ' "I] = (-8; 20; -16) = -4(2; -5; 4) Vay p h u o n g trinh ( Q ) : 2x - 5y + 4z + 3 = 0. 3) Cach 1: Giai bang p h u o n g phap dai so
V i (P) d i qua M nen p h u o n g trinh (P) c6 dang: a ( x - l ) + b ( y - 2 ) + c ( z - l ) = : 0 voi a ^ + b ^ + c ^ > 0