If the shortest distance between the lines x - 4 1 = y + 1 2 = z - 3 and x - λ 2 = y + 1 4 = z - 2 - 5 is 6 5 , then the sum of all possible values of λ is :

Step-by-step solution

x - 4 1 = y + 1 2 = z - 3 x - λ 2 = y + 1 4 = z - 2 - 5 The shortest distance between the lines = ( a → - b → ) · d 1 → × d 2 → d 1 → × d 2 → a ≡ 4 , - 1 , 0 , b ≡ λ , - 1 , 2 ⇒ a → - b → = 4 i ^ - j ^ - λ i ^ - j ^ + 2 k ^ ⇒ a → - b → = 4 - λ i ^ - 2 k ^ So, the shortest distance is given by, D = λ - 4 0 2 1 2 - 3 2 4 - 5 i ^ j ^ k ^ 1 2 - 3 2 4 - 5 ⇒ D = ( λ - 4 ) ( - 10 + 12 ) - 0 + 2 ( 4 - 4 ) | 2 i ^ - 1 j ^ + 0 k ^ | ⇒ 6 5 = 2 ( λ - 4 ) 5 ⇒ 3 = | λ - 4 | ⇒ λ - 4 = ± 3 ⇒ λ = 7 , 1 So, sum of all possible value