A line with direction cosines proportional to 2, 1, 1 meets each of the lines x=y+a=z  andx+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 (λ,λ−a,λ) and a point on the line x+a=2y=2z is (μ−a,μ2,μ2) ,  then direction ratio of the line joining these points are λ−μ+a,λ−a−μ2,λ−μ2 If it represents the required line, then λ−μ+a2=λ−a−μ21=λ−μ22 On solving we get λ=3a,μ=2a ∴ the required points of intersection are (3a,3a−a,3a) and (2a−a,2a2,2a2) or (3a,2a,3a)  and (a,a,a)