A line with direction cosines proportional to 2, 1, 1 meets each of the lines _$x=y+a=z$  and_$x+a=2y=2z$ . The coordinates of each of the points of intersection are given by

Let a point on the line _$x=y+a=z$ is _{$\left(\lambda ,\lambda -a,\lambda \right)$} and a point on the line _$x+a=2y=2z$ is _{$\left(\mu -a,\frac{\mu }{2},\frac{\mu }{2}\right)$} ,

 then direction ratio of the line joining these points are _{$\lambda -\mu +a,\lambda -a-\frac{\mu }{2},\lambda -\frac{\mu }{2}$}

If it represents the required line, then _{$\frac{\lambda -\mu +a}{2}=\frac{\lambda -a-\frac{\mu }{2}}{1}=\frac{\lambda -\frac{\mu }{2}}{2}$}

On solving we get _{$\lambda =3a,\mu =2a$}

_$\therefore$ the required points of intersection are _{$\left(3a,3a-a,3a\right)$} and _{$\left(2a-a,\frac{2a}{2},\frac{2a}{2}\right)$} or _{$\left(3a,2a,3a\right)$}  and _{$\left(a,a,a\right)$}