S o l u t i o n Solution S o l u t i o n
We will first find the distance between the two lines. Then that is the diameter of smallest sphere touching the two st. lines. Dividing the diameter by 2 we will get the radius of the sphere.
The two lines L 1 L_1 L 1 and L 2 L_2 L 2 are
< t 1 − 2 , 2 − 2 t 1 , t 1 − 6 > a n d < 7 t 2 + 21 , − 6 t 2 − 18 , t 2 + 3 > <t_1-2,\ 2-2t_1,\ t_1-6>\ and\ <7t_2+21,\ -6t_2-18,\ t_2+3> < t 1 − 2 , 2 − 2 t 1 , t 1 − 6 > an d < 7 t 2 + 21 , − 6 t 2 − 18 , t 2 + 3 > The vector P ⃗ \vec{P} P between the two touchpoints P 1 P _1 P 1 and P 2 P _2 P 2 must be perpendicular to both direction vectors of the lines L 1 a n d L 2 L_1\ and\ L_2 L 1 an d L 2
P ⃗ = P 1 − P 2 = [ ( t 1 − 2 ) − ( 7 t 2 + 21 ) ( − 2 t 1 + 2 ) − ( − 6 t 2 − 18 ) ( t 1 − 6 ) − ( t 2 + 3 ) ] ⟹ [ t 1 − 7 t 2 − 23 − 2 t 1 + 6 t 2 + 20 t 1 − t 2 − 9 ] P ⃗ [ 1 − 2 1 ] = ( t 1 − 7 t 2 − 23 ) − 2 ( − 2 t 1 + 6 t 2 + 20 ) + ( t 1 − t 2 − 9 ) = 0 ⟹ 6 t 1 − 20 t 2 − 72 = 0 P ⃗ [ 7 − 6 1 ] = − 7 ( t 1 − 7 t 2 − 23 ) + 6 ( − 2 t 1 + 6 t 2 + 20 ) + ( t 1 − t 2 − 9 ) = 0 ⟹ − 20 t 1 + 86 t 2 + 290 = 0 \vec{P}=P_1-P_2=
\begin{bmatrix}
(t_1-2)-(7t_2+21)\\
(-2t_1+2)-(-6t_2-18)\\
(t_1-6)-(t_2+3)
\end{bmatrix} \implies
\begin{bmatrix}
t_1-7t_2-23\\
-2t_1+6t_2+20\\
t_1-t_2-9
\end{bmatrix}\\
\vec{P}\begin{bmatrix}1\\ -2\\ 1 \end{bmatrix}=(t_1-7t_2-23)-2(-2t_1+6t_2+20)+(t_1-t_2-9)=0\\
\implies 6t_1-20t_2-72=0\\
\vec{P}\begin{bmatrix}7\\ -6\\ 1 \end{bmatrix}=-7(t_1-7t_2-23)+6(-2t_1+6t_2+20)+(t_1-t_2-9)=0\\
\implies -20t_1+86t_2+290=0\\ P = P 1 − P 2 = ⎣ ⎡ ( t 1 − 2 ) − ( 7 t 2 + 21 ) ( − 2 t 1 + 2 ) − ( − 6 t 2 − 18 ) ( t 1 − 6 ) − ( t 2 + 3 ) ⎦ ⎤ ⟹ ⎣ ⎡ t 1 − 7 t 2 − 23 − 2 t 1 + 6 t 2 + 20 t 1 − t 2 − 9 ⎦ ⎤ P ⎣ ⎡ 1 − 2 1 ⎦ ⎤ = ( t 1 − 7 t 2 − 23 ) − 2 ( − 2 t 1 + 6 t 2 + 20 ) + ( t 1 − t 2 − 9 ) = 0 ⟹ 6 t 1 − 20 t 2 − 72 = 0 P ⎣ ⎡ 7 − 6 1 ⎦ ⎤ = − 7 ( t 1 − 7 t 2 − 23 ) + 6 ( − 2 t 1 + 6 t 2 + 20 ) + ( t 1 − t 2 − 9 ) = 0 ⟹ − 20 t 1 + 86 t 2 + 290 = 0
Solving simultaneously, we get
t 1 = 98 29 t 2 = − 75 29 P 1 = { 40 29 − 138 29 , − 76 29 } P 2 = { 84 29 − 72 29 , 12 29 } t_1=\frac{98}{29}\\
t_2=-\frac{75}{29}\\
P_1=\{\frac{40}{29}\, -\frac{138}{29}, \ -\frac{76}{29} \}\\
P_2=\{\frac{84}{29}\, -\frac{72}{29}, \ \frac{12}{29} \}\\ t 1 = 29 98 t 2 = − 29 75 P 1 = { 29 40 − 29 138 , − 29 76 } P 2 = { 29 84 − 29 72 , 29 12 }
The center of the sphere is the middle between the touchpoints:
M = { 62 29 , − 105 29 , − 32 29 , } r 2 = ∣ P ⃗ ∣ 4 = ( t 1 − 7 t 2 − 23 ) 2 + ( − 2 t 1 + 6 t 2 + 20 ) 2 + ( t 1 − t 2 − 9 ) 2 4 = 121 29 M=\{\frac{62}{29},\ -\frac{105}{29},\ -\frac{32}{29},\ \}\\
r^2=\frac{|\vec{P}|}{4}=\frac{(t_1-7t_2-23)^2+(-2t_1+6t_2+20)^2+(t_1-t_2-9)^2}{4}=\frac{121}{29} M = { 29 62 , − 29 105 , − 29 32 , } r 2 = 4 ∣ P ∣ = 4 ( t 1 − 7 t 2 − 23 ) 2 + ( − 2 t 1 + 6 t 2 + 20 ) 2 + ( t 1 − t 2 − 9 ) 2 = 29 121
Thus the smallest sphere is the sphere with the equation
( x − 62 29 ) 2 + ( y + 105 29 ) 2 + ( z + 32 29 ) 2 = 121 29 − − − > a n s w e r (x-\frac{62}{29})^2+(y+\frac{105}{29})^2+(z+\frac{32}{29})^2=\frac{121}{29}--->answer ( x − 29 62 ) 2 + ( y + 29 105 ) 2 + ( z + 29 32 ) 2 = 29 121 − − − > an s w er
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