If the shortest distance between the lines
(x-λ)/-2=(y-2)/1=(z-1)/1 and (x-√3)/1=(y-1)/-2=(z-2)/1 is 1 , then the sum of all possible values of λ is :

Solution

Maths sir · 4 steps

Cross the two directions to get the common normal, then project the joining vector onto it.

1. Read off the lines. Line 1 passes through A(λ, 2, 1) with direction b₁⃗ = (-2, 1, 1). Line 2 passes through B(√3, 1, 2) with direction b₂⃗ = (1, -2, 1).
2. Common normal. b₁⃗ × b₂⃗ = (1 · 1 - 1 · (-2), 1 · 1 - (-2) · 1, (-2)(-2) - 1 · 1) = (3, 3, 3). Its length is 3√3. Also AB⃗ = (√3 - λ, -1, 1).

   d = (|AB⃗ · (b₁⃗ × b₂⃗)|)/(|b₁⃗ × b₂⃗|) = |3(√3 - λ - 1 + 1)|/3√3 = (|√3 - λ|)/√3

3. Equate to 1. |λ - √3| = √3, so λ - √3 = √3 or λ - √3 = -√3. That gives λ = 2√3 or λ = 0.
4. So. The sum of all possible values is 2√3 + 0 = 2√3, option (b).

The idea to keep. Shortest distance between skew lines is the projection of any joining vector on b₁⃗ × b₂⃗. The modulus gives two values of the unknown, so keep both.

Answer from the NTA final answer key. Solution by PrepWiser.