Let the line of the shortest distance between the lines L 1 : r → = i ^ + 2 j ^ + 3 k ^ + λ i ^ − j ^ + k ^ and L 2 : r → = 4 i ^ + 5 j ^ + 6 k ^ + μ i ^ + j ^ − k ^ intersect L 1 and L 2 at P and Q respectively. If α , β , γ is the midpoint of the line segment P Q , then 2 α + β + γ is equal to___________

Step-by-step solution

Let point on L 1 be P 1 + λ i ^ + 2 - λ j ^ + 3 + λ k ^ and point on L 2 be Q 4 + μ i ^ + 5 + μ j ^ + 6 - μ k ^ . ⇒ P Q → = λ - μ - 3 i ^ + - λ - μ - 3 j ^ + λ + μ - 3 k ^ Now, P Q → ⊥ L 1 , L 2 ⇒ λ - μ - 3 + λ + μ + 3 + λ + μ - 3 = 0 ⇒ 3 λ + μ = 3 . . . i Also, λ - μ - 3 + - λ - μ - 3 + - λ - μ + 3 = 0 ⇒ - λ - 3 μ = 3 . . . i i Now, solving above equations we get, ⇒ 3 λ + μ - 3 λ - 9 μ = 3 + 9 ⇒ - 8 μ = 12 ⇒ μ = - 3 2 ⇒ λ = 3 2 ⇒ P ≡ 5 2 , 1 2 , 9 2 , Q ≡ 5 2 , 7 2 , 15 2 So, mid-point of P Q is 5 2 , 2 , 6 ⇒ 2 α