If the lines 2x = ky = -z and 6x = -y = -4z are perpendicular to each other then the value of k is ...

Step-by-step solution

The equations of the given lines can be rewritten in standard symmetric form as: x 1/2 = y 1/k = z -1 x 1/6 = y -1 = z -1/4 The direction ratios of the first line are a₁ = 1 2 , b₁ = 1 k , c₁ = -1 . The direction ratios of the second line are a₂ = 1 6 , b₂ = -1 , c₂ = - 1 4 . Since the two lines are perpendicular to each other, the sum of the products of their corresponding direction ratios must be zero: a₁ a₂ + b₁ b₂ + c₁ c₂ = 0 ( 1 2 ) ( 1 6 ) + ( 1 k )(-1) + (-1) (- 1 4 ) = 0 1 12 - 1 k + 1 4 = 0 1 k = 1 12 + 3