If the distance between the plane, 23 x - 10 y - 2 z + 48 = 0 and the plane containing the lines x + 1 2 = y - 3 4 = z + 1 3 and x + 3 2 = y + 2 6 = z - 1 λ λ ∈ R is equal to k 633 , then k is equal to ____________.

Step-by-step solution

Lines must be intersecting ⇒ 2 s - 1 , 4 s + 3 , 3 s - 1 = 2 t - 3 , 6 t - 2 , λ t + 1 2 s - 1 = 2 t - 3 , 4 s + 3 = 6 t - 2 , 3 s - 1 = λ t + 1 ⇒ t = 1 2 , s = - 1 2 , λ = - 7 Distance of plane contains given lines from given plane is same as distance between point - 3 , - 2 , 1 from given plane. Required distance equal to - 69 + 20 - 2 + 48 529 + 100 + 4 = 3 633 = k 633 ⇒ k = 3