Let A and B be two distinct points on the line L: (x-6)/(3)=(y-7)/(2)=(z-7)/(-2).

Question

Let $A$ and $B$ be two distinct points on the line $L: \frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}$. Both $A$ and $B$ are at a distance $2 \sqrt{17}$ from the foot of perpendicular drawn from the point $(1,2,3)$ on the line $L$. If $O$ is th...

Accepted Answer

$L:\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}$

Point $A(3\lambda +6,2\lambda +7,7-2\lambda )$

$B(3\mu +6,2\mu +7,7-2\mu )$

Let $P(3k+6,2k+7,7-2k)$ be foot of perpendicular from ${P}^{'}(1,2,3)$

$\begin{gathered} 	herefore \, P{P}^{'}\langle 3,2,-2angle =0 \ 3(3k+5)+(2k+5)2+(4-2k)(-2)=0 \ 9k+15+4k+10-8+4k=0 \ 17k+17=0 \ \Rightarrow k=-1\, 	herefore \, P(3,5,9) \ |\overset{	o \leftarrow }{AP}|=2\sqrt{17} \end{gathered}$

$\begin{gathered} \Rightarrow (3\lambda +3{)}^{2}+(2\lambda +2{)}^{2}+(-2-2\lambda {)}^{2}=17	imes 4 \ =17(\lambda +1{)}^{2}=17	imes 4 \ \Rightarrow \lambda +1=\pm 2\Rightarrow \lambda =1\lambda =-3 \ 	herefore A(9,9,5),B(-3,1,13) \ \overset{	o \leftarrow }{OA}\cdot \overset{	o \leftarrow }{OB}=(9\hat{i}+9\hat{j}+5\hat{k})\cdot (-3\hat{i}+\hat{j}+13\hat{k}) \ =-27+9+65=47 \end{gathered}$

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