Question #133321

find the smallest sphere which touches the lines (x-2)/1=(y-1)/-2=(z-6)/1 and (x+3)/7=(y+3)/-6=(z+3)/1 


1
Expert's answer
2020-09-20T17:51:37-0400
SolutionSolution

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 L1L_1 and L2L_2 are

<t12, 22t1, t16> and <7t2+21, 6t218, t2+3><t_1-2,\ 2-2t_1,\ t_1-6>\ and\ <7t_2+21,\ -6t_2-18,\ t_2+3>

The vector P\vec{P} between the two touchpoints P1P _1 and P2P _2 must be perpendicular to both direction vectors of the lines L1 and L2L_1\ and\ L_2


P=P1P2=[(t12)(7t2+21)(2t1+2)(6t218)(t16)(t2+3)]    [t17t2232t1+6t2+20t1t29]P[121]=(t17t223)2(2t1+6t2+20)+(t1t29)=0    6t120t272=0P[761]=7(t17t223)+6(2t1+6t2+20)+(t1t29)=0    20t1+86t2+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\\

Solving simultaneously, we get

t1=9829t2=7529P1={402913829, 7629}P2={84297229, 1229}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} \}\\

The center of the sphere is the middle between the touchpoints:


M={6229, 10529, 3229, }r2=P4=(t17t223)2+(2t1+6t2+20)2+(t1t29)24=12129M=\{\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}


Thus the smallest sphere is the sphere with the equation


(x6229)2+(y+10529)2+(z+3229)2=12129>answer(x-\frac{62}{29})^2+(y+\frac{105}{29})^2+(z+\frac{32}{29})^2=\frac{121}{29}--->answer


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