$\text{ The intersection of the planes }2x-y-3z=8\text{ and }x+2y-4z=14\text{ is the line }L.\text{ The value of '}a\text{'}\text{ for which }$

$\text{ the line }L\text{ is perpendicular to the line through }(a,2,2)\text{ and }$$(6,11,-1)\text{ is }$

Vector along the line of intersection of planes is
$\left|\begin{array}{ccc}\widehat{i}& \widehat{j}& \widehat{k}\\ 2& -1& -3\\ 1& 2& -4\end{array}\right|=10\widehat{i}+5\widehat{j}+5\widehat{k}$

This vector is perpendicular to the segment joining the points (a, 2, 2) and (6, 11, -1). 

$\begin{array}{l}\therefore 10(a-6)+5(2-11)+5(2+1)=0\\ \Rightarrow a=9\end{array}$