duong tron c6 chu vi bang 8n
3) Tim m de mat cau ( S ) cat duong th^ng A tai hai diem M , N sao cho
M N = 8 . ,
JCgigidị
Mat cau (S) c6 tarn I ( - l ; 2 ; 3 ) va ban kinh R = V l 7 - m voi m < 17. 1) GQ'I (P) la mat phSng di qua A va Ạ
Ta CO A di qua B(l;-1;3), c6 VTCP u -(2;2;1) Suy ra AB = (-2;-3;2), n^ = AB A U = (-7;6;2). Phuong trinh (P): 7x - 6y - 2z - 7 = 0.
Do do (S) tiep xiic voi (P) khi va chi khi d(I,(P)) = R /— 32 489
o V l 7 - m = - 7 =/89 89 • <=> m = •
2) Goi r la ban kinh cua duong tron la giao tuyen cua (S) voi (P) Ta c6: 27ir = 871 r : ^ r = 4, d(I,(P)) = 2. Vi R2 =r2 +d2(I,(P)) nentaco: 1 7 - m = 4^ +2^ o m - - 3 . 3) Ta CO IB = (2; -3; 0) =i> IB A u = (-3; -2; 10) IB A u 7Ti3 Do do d ( I, A ) - , MKl2 T 113 104 Vay R 2 = i ^ + d 2( I, A) o l 7 - m = 16 + — « m = - — 281
I'hinniiy filtdp gidi Todn Hinh hoc theo chuyen ite - Nguyen Phu Klidnh, Ngut/en Tat Thu
Vi 3.4.5. Trong khong gian Oxyz cho mat phang (a): 2x + y + 2z - 1 = 0 va fx = - l + 2t
hai duang thang Aj : y = -2 + t va A2 : x - 3 _ y _ z - 5
1
z = -2 + 2t
1) Viet phuang trinh mat cau (S) c6 tam nam tren duang thang A j , tiep xiic vai duang thang Aj va (a).
2) Viet phuong trinh mat cau (S) c6 tam nam tren A j , tiep xiic vai (a) va cat duang thang Aj tai hai diem M , N sao cho M N = 2\/30
3) Viet phuang trinh mat cau (S) c6 tam nam tren A j , tiep xiic vai Aj va cat (a) theo mot duang tron c6 ban kinh bang Vis.
Xffi gidị
1) Gpi I la tam mat cau, R la ban kinh. Do I e A2 nen ta c6: 1(3 + 1 ; - t ; 5 +1)
Duong thang Aj di qua M j ( - l ; - 2 ; - 2 ) , VTCP \x[ = {2;V,1)
Suy ra M ^ I = (t + 4;-t + 2;t + 7), M J I A U J - ( - 3 t - 3 ; 6 ; 3 t ) • = >/(t + l)2+4 + t2 =\/2t2+2t + 5 |M|I A U j
Dodo: d(I,Aj) =
Vi (S) tiep xiic voi Aj va (a) nen ta c6: d(I,(a)) = d(I,Aj) = R t + 5 2t^+2t + 5 o t ^ - 8 t - 2 0 - 0 o t = 10,t = -2 • t = - 2 , s u y ra I(l;2;3), R = 3.
Phuang trinh (S): (x -1)^ + (y - 2)^ + (z - 3)^ = 9 . • t = 10,suyra I(13;-10;15), R = 15.
Phuang trinh (S): (x -13)^ + (y +10)^ + (z -15)^ = 225. 2) Goi I la tam mat cau, suy ra I ( - l + 2t;-2 +1;-2 + 2t)
Duang thang ^2 qua M2(3;0;5), c6 VTCP u^ = ( l ; - l ; l ) Suy ra = ( 2 t - 4 ; t - 2 ; 2 t - 7 ) , M 2 I A U 2 = ( 3 t - 9 ; - 3 ; - 3 t + 6) Nen d(I,A2) = ^ 3 [ ( t - 3) 2+ l + ( t - 2) 2 ] = ^?,{2i^ - l O t + 14) Vi mat cau tiep xiic vai (a) nen R = d(I,(a)) = 3t - 3 Tudangthiic R^ = ^ - ^ + d^(I,A2) ta c6:
'82 Cty TNHH MTV D W H KhangVỉl (3t - 3)2 = 30 + 3{2t^ - lot +14) <=> t^ + 4t - 21 = 0 <=> t = -7, t = 3 t = 3, suy ra I(5;l;4), R = 6. Phuang trinh (S): (x - 5)^ + (y -1)^ + (z - 4f = 36. ^ t = - 7 , s u y ra I(-15;-9;-16), R = 24. K Phuang trinh (S): (x +15)^ + (y + 9f + (z +16)^ = 576.
H:GQi I la tam mat cau, suy ra 1(3 +1; - t ; 5 +1) ^
Taco: R = d(I,Ai) = iMjI A U j - J 2t^+2t + 5
^ Vi (S) cat mat phang (a) theo duang tron c6 ban kinh r = y/l3 nen ta c6: • R^ = r^ + d^{l,{a)) o 2t2 + 2t + 5 = 13 + (t + 5)^
W <=>t^-8t-33 = 0<=>t = -3,t = l l .
M t = - 3 , s u y ra I(0;3;2), R = Vl7
H p h u a n g trinh (S): x^ + (y - 3)^ + (z - if = 17 .
IP t = 11, suy ra I(14;-ll;16), R = V269
Phuang trinh (S):(x-14)2+(y + l l) 2+ ( z - 1 6 ) ^ =269.
Vi du 3.4.6. Trong khong gian Oxyz, cho Ăa;0;0), B(0;b;0), C(0;0;c)vdi
a, b, c > 0 v a i + - + - = 2. a b e