A line L with direction ratios 2, -1, 1 intersects the lines x - 1 = y - 1 2 = z - 1 and x - 3 2 = y = z - 2 3 at points P and Q respectively. If the line L also intersects the plane x + y + z = 7 at a point R , then the square of the distance of R from the point (1, 2, 3) is

Step-by-step solution

Let the general points on the given lines be P(t+1, 2t+1, t+1) and Q(2s+3, s, 3s+2) . The direction ratios of the line PQ are (2s - t + 2, s - 2t - 1, 3s - t + 1) . Since the direction ratios of L are given as 2, -1, 1 , we have: 2s - t + 2 2 = s - 2t - 1 -1 = 3s - t + 1 1 Taking the first two parts: -2s + t - 2 = 2s - 4t - 2 4s - 5t = 0 Taking the last two parts: s - 2t - 1 = -3s + t - 1 4s - 3t = 0 Solving these equations yields s = 0 and t = 0 . Thus, the points are P(1, 1, 1) and Q(3, 0, 2) . The equation of th