Let S be the set of all values of λ, for which the shortest distance between
the lines (x-λ)/0=(y-3)/4=(z+6)/1 and (x+λ)/3=y/-4=(z-6)/0 is 13. Then 8|Σ_(λ ∈ S) λ| is equal to :

Solution

Maths sir · 4 steps

Take one point and one direction from each line; the shortest distance formula gives two values of λ.

1. Read off the lines. Line 1 passes through A(λ, 3, -6) with direction b⃗₁ = (0, 4, 1). Line 2 passes through B(-λ, 0, 6) with direction b⃗₂ = (3, -4, 0).
2. Cross product. b⃗₁ × b⃗₂ = (4 · 0 - 1 · (-4), 1 · 3 - 0 · 0, 0 · (-4) - 4 · 3) = (4, 3, -12). Its magnitude is √(16 + 9 + 144) = 13.

   d = (|(B⃗-A⃗) · (b⃗₁ × b⃗₂)|)/(|b⃗₁ × b⃗₂|) = (|(-2λ, -3, 12) · (4, 3, -12)|)/13 = (|8λ + 153|)/13

3. Set it to 13. |8λ + 153| = 169, so 8λ + 153 = 169 or 8λ + 153 = -169. That gives 8λ = 16 or 8λ = -322, so λ = 2 or λ = -161/4.
4. So. 8|Σ λ| = |8λ₁ + 8λ₂| = |16 - 322| = 306, which is option (a).

The idea to keep. With a modulus equation like |8λ + c| = k, both signs give a valid λ. Keep both, and when asked for 8Σλ, add the 8λ values directly.

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