If the lines x - 1 2 = y - 1 = z 2 and x - y + z - 2 = 0 = λ x + 3 z + 5 are coplanar, then the value of 7 λ is equal to

Step-by-step solution

A plane passing through the line of intersection of the given 2 planes is x - y + z - 2 + μ λ x + 3 z + 5 = 0 ⇒ x 1 + λ μ - y + z 1 + 3 μ + 5 μ - 2 = 0 Since the given line x - 1 2 = y - 1 = z 2 must lie in this plane ⇒ 2 1 + λ μ + - 1 - 1 + 1 + 3 μ 2 = 0 ⇒ 2 + 2 λ μ + 1 + 2 + 6 μ = 0 ⇒ 2 λ μ + 6 μ + 5 = 0 … . . i The point 1 , 0 , 0 must satisfy the equation of the plane ⇒ 1 + λ μ - 0 + 0 + 5 μ - 2 = 0 ⇒ λ μ + 5 μ - 1 = 0 … … i i Solving i and i i we get, μ = 7 4 , λ = - 31 7 ⇒ 7 λ = - 31