Question

Let $\lambda$ be an integer. If the shortest distance between the lines $x-\lambda =2y-1=-2z$ and $x=y+2\lambda =z-\lambda$ is $\frac{\sqrt{7}}{2\sqrt{2}},$ then the value of $|\lambda |$ is

Answer 1

$\frac{x-\lambda }{1}=\frac{y-\frac{1}{2}}{\frac{1}{2}}=\frac{z}{-\frac{1}{2}}$
or, $\frac{x-\lambda }{2}=\frac{y-\frac{1}{2}}{1}=\frac{z}{-1}\cdots \left(1\right)$
Point on line $\left(1\right)$ is $(\lambda ,\frac{1}{2},0)$

$\frac{x}{1}=\frac{y+2\lambda }{1}=\frac{z-\lambda }{1}\cdots \left(2\right)$
Point on line $\left(2\right)$ is $(0,-2\lambda ,\lambda )$

Distance between skew lines
$=\frac{[\overrightarrow{a_2}-\overrightarrow{a_1}\overrightarrow{b_1}\overrightarrow{b_2}]}{|\overrightarrow{b_1}\times \overrightarrow{b_2}|}$
$=\frac{\left|\begin{array}{ccc}\lambda & \frac{1}{2}+2\lambda & -\lambda \\ 2& 1& -1\\ 1& 1& 1\end{array}\right|}{\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& 1& -1\\ 1& 1& 1\end{array}\right|}$
$=\frac{|-5\lambda -\frac{3}{2}|}{\sqrt{14}}=\frac{\sqrt{7}}{2\sqrt{2}}$ (Given)
$\Rightarrow |10\lambda +3|=7$
$\Rightarrow \lambda =-1$ as $\lambda$ is an integer.
$\Rightarrow |\lambda |=1$