The lines x - 2 1 = y - 3 1 = z - 4 -k and x - 1 k = y - 4 2 = z - 5 1 are coplanar if

Step-by-step solution

The condition for two lines x - x₁ a₁ = y - y₁ b₁ = z - z₁ c₁ and x - x₂ a₂ = y - y₂ b₂ = z - z₂ c₂ to be coplanar is: vmatrix x₂ - x₁ & y₂ - y₁ & z₂ - z₁ a₁ & b₁ & c₁ a₂ & b₂ & c₂ vmatrix = 0 From the given equations of the lines, we have (x₁, y₁, z₁) = (2, 3, 4) and (x₂, y₂, z₂) = (1, 4, 5) . The direction ratios are (a₁, b₁, c₁) = (1, 1, -k) and (a₂, b₂, c₂) = (k, 2, 1) . Substituting these values into the determinant condition: vmatrix 1 - 2 & 4 - 3 & 5 - 4 1 & 1 & -k k & 2 & 1 vmatrix = 0 vmatrix -1 & 1 & 1 1