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 :

Solution

Maths sir · 4 steps

The modulus in the shortest distance formula gives two values of λ, so add them both.

1. Read off the lines. Line 1 passes through A(4, -1, 0) with direction b⃗₁ = (1, 2, -3). Line 2 passes through B(λ, -1, 2) with direction b⃗₂ = (2, 4, -5).
2. Cross product. b⃗₁ × b⃗₂ = (2 × (-5) - (-3) × 4, (-3) × 2 - 1 × (-5), 1 × 4 - 2 × 2) = (2, -1, 0), with magnitude √5. Also AB⃗ = (λ - 4, 0, 2).

   d = (|AB⃗ · (b⃗₁ × b⃗₂)|)/(|b⃗₁ × b⃗₂|) = |2(λ - 4)|/√5

3. Set it equal. |2(λ - 4)|/√5 = 6/√5 gives |λ - 4| = 3. So λ - 4 = 3 or λ - 4 = -3, that is λ = 7 or λ = 1.
4. So. The sum of all possible values is 7 + 1 = 8, option (d).

The idea to keep. Shortest distance has a modulus in the numerator. When the unknown sits inside it, you always get two values, one from each sign.

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