If the shortest between the lines (x + √6)/2 = (y - √6)/3 = (z - √6)/4 and (x - λ)/3 = (y - 2√6)/4 = (z + 2√6)/5 is 6, then the square of sum of all possible values of λ is :

Solution

Maths sir · 4 steps

Use the cross product formula for shortest distance, and keep the modulus: it gives you two values of λ.

1. Read the lines. Line 1 passes through A(-√6, √6, √6) with direction b⃗₁ = (2, 3, 4). Line 2 passes through B(λ, 2√6, -2√6) with direction b⃗₂ = (3, 4, 5).
2. Cross product. b⃗₁ × b⃗₂ = (15 - 16, 12 - 10, 8 - 9) = (-1, 2, -1), with magnitude √6. Also AB⃗ = (λ + √6, √6, -3√6).

   d = (|AB⃗·(b⃗₁ × b⃗₂)|)/(|b⃗₁ × b⃗₂|) = (|-(λ + √6) + 2√6 + 3√6|)/√6 = (|4√6 - λ|)/√6

3. Set it to 6. |λ - 4√6| = 6√6, so λ - 4√6 = ± 6√6. This gives λ = 10√6 or λ = -2√6.
4. So. The sum of the values is 8√6, and its square is 64 × 6 = 384.

The common slip: 600. Dropping the modulus keeps only λ = 10√6, and squaring that gives 600 instead of 384.

The idea to keep. Shortest distance between skew lines is the projection of the joining vector on b⃗₁ × b⃗₂. A modulus equation always gives two cases, so check both.

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