Question

A line whose direction cosines is proportional to $2,1,2$, meets with the lines $x=y+a=z$ and $x+a=2y=2z$ at $A$ and $B$ respectively. If the distance between $A$ and $B$ is $d$, then $\frac{12d}{|a|}$ is

Answer 1

For $A$,
let $x=y+a=z=t$
$\Rightarrow A\equiv (t,t-a,t)$
For $B$,
let $\frac{x+a}{2}=y=z=k$
$\Rightarrow B\equiv (2k-a,k,k)$

Now, the direction ratios of $AB$ are $(t+a-2k,t-a-k,t-k)$
Since, direction ratios are parallel to $(2,1,2),$
$\frac{t+a-2k}{2}=\frac{t-a-k}{1}=\frac{t-k}{2}$
$\Rightarrow t=3a,k=a$
$\therefore A\equiv (3a,2a,3a),B\equiv (a,a,a)$
$\Rightarrow d=3|a|$