Show that the two lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$    and  $\frac{x-4}{5}=\frac{y-1}{2}=z$ , intersect. Also find the point of intersection of these lines

Here,    $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}\mathrm{...}\left(i\right)$
And  $\frac{x-4}{5}=\frac{y-1}{2}=\frac{z-0}{1}\mathrm{...}\left(ii\right)$
Any point on line (i) is $P(2r+1,3r+2,4r+3)$   and any point on the line (ii) is  $Q(5\lambda +4,2\lambda +1,\lambda )$
They intersect if and only if  $2r+1=5\lambda +4,3r+2=2\lambda +1,4r+3=\lambda$
On solving,  $r=-1,\lambda =-1$
Clearly, for these values of  $\lambda$   and  $rP(-1,-1,-1)$
Hence, lines (i) and (ii) intersect at  $(-1,-1,-1)$