A line with direction ratios 1, -1, 2 intersects the lines x 2 = y 3 = z+1 3 and x+1 -1 = y-2 1 = z 4 at the points P and Q , respectively. If the length of the line segment PQ is , then 225 ^2 is equal to:

Step-by-step solution

Let the coordinates of P on the first line be (2 , 3 , 3 - 1) . Let the coordinates of Q on the second line be (- - 1, + 2, 4 ) . The direction ratios of the line segment PQ are proportional to 1, -1, 2 . Thus, we can write: - - 1 - 2 1 = + 2 - 3 -1 = 4 - 3 + 1 2 = k This gives the system of equations: - - 2 - 1 = k - 3 + 2 = -k 4 - 3 + 1 = 2k Adding the first two equations gives: -5 + 1 = 0 = 1 5 Substituting = 1 5 into the first and third equations: - - 7 5 = k + k = - 7 5 4 + 2 5 = 2k 2 - k = - 1 5 Adding these