A line with direction cosines proportional to 2, 1, 2 meets each of the lines x = y + a = z and x + a = 2y = 2z. The co-ordinates of each of the points of intersection are given by

$(3a, 2a, 3a), (a, a, a)$

We observe that the points $P(3a, 2a, 3a)$ and $Q(a, a, a)$ satisfy the equations of the lines $x = y + a = z$ and $x + a = 2y = 2z$ respectively. Also, direction ratios of PQ are proportional to $3a - a, 2a - a, 3a - a$ i.e., 2, 1, 2.

Hence, option (b) is correct.