If two lines L₁: x-1 2 = y+1 3 = z-1 4 and L₂: x-3 1 = y-k 2 =z intersect at a point, then 2 k is equal to

Step-by-step solution

The lines are given by L₁: x-1 2 = y+1 3 = z-1 4 = and L₂: x-3 1 = y-k 2 = z 1 = . Any point on L₁ is (2 + 1, 3 - 1, 4 + 1) and any point on L₂ is ( + 3, 2 + k, ) . Since the lines intersect, there exist and such that: 2 + 1 = + 3 2 - = 2 (1) 3 - 1 = 2 + k 3 - 2 = k + 1 (2) 4 + 1 = (3) Substituting (3) into (1): 2 - (4 + 1) = 2 -2 = 3 = - 3 2 . From (3), = 4(- 3 2 ) + 1 = -6 + 1 = -5 . Substituting = - 3 2 and = -5 into (2): 3(- 3 2 ) - 2(-5) = k + 1 - 9 2 + 10 = k + 1 11 2 = k + 1 k = 9 2 . Therefore, 2k = 2 9 2 =