If the two lines  $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and  $\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}$ have a point in common, then k = 

$\text{ Givewn lines are }\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}=\lambda$
Let $\mathrm{P}=(2\lambda +1,3\lambda -1,4\lambda +1)$ be any point on this line
$\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}=\mu$
$\text{ Let }\mathrm{Q}=(\mu +3,2\mu +\mathrm{k},\mu )\text{ be any point on this line }$
$\begin{array}{r}2\lambda +1=\mu +3\Rightarrow 2\lambda -\mu -2=0\dots \left(1\right)\\ 3\lambda -1=2\mu +\mathrm{k}\Rightarrow 3\lambda -2\mu -1-\mathrm{k}=0\dots \left(2\right)\end{array}$
$1+4\lambda =\mu$
$\begin{array}{r}\text{ from (1) }2\lambda -(1+4\lambda )-2=0\Rightarrow \lambda =\frac{-3}{2}\\ \text{ and }\mu =1+4\left(\frac{-3}{2}\right)=-5\\ \text{ from (2) }k=\frac{-9}{2}+10-1=\frac{9}{2}\end{array}$