If the lines _{$\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$} and _{$\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}$} intersect, then the value of k is

_{$\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ =$\lambda$ $\Rightarrow x=2\lambda +1,y=3\lambda -1\&z=4\lambda +1$}

_{$\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}$ $=\mu$ $\Rightarrow x=3+\mu ,y=k+2\mu \&z=\mu$}

Since above line intersect _{$\Rightarrow 2\lambda +1=3+\mu \mathrm{.............}\left(1\right)$}

_{$3\lambda -1=2\mu +k\mathrm{...............}\left(2\right)$}

_{$\mu =4\lambda +\mathrm{1..............}\left(3\right)$}

Solving (1) & (3) and put the value of _{$\lambda \&\mu$} in (2) _{$\Rightarrow k=\frac{9}{2}$}