If the shortest distance between the lines (x - 1)/2 = (y - 2)/3 = (z - 3)/λ and (x - 2)/1 = (y - 4)/4 = (z - 5)/5 is 1/√3, then the sum of all possible value of λ is :

Solution

Maths sir · 4 steps

Write the distance as a function of λ, set it equal to 1/√3, and read the sum of roots off the quadratic.

1. Read off the lines. Line 1 passes through A(1, 2, 3) with direction b⃗₁ = (2, 3, λ). Line 2 passes through B(2, 4, 5) with direction b⃗₂ = (1, 4, 5). So AB⃗ = (1, 2, 2).

   d = (|AB⃗ · (b⃗₁ × b⃗₂)|)/(|b⃗₁ × b⃗₂|)

2. Cross and dot. b⃗₁ × b⃗₂ = (15 - 4λ, λ - 10, 5), so |b⃗₁ × b⃗₂|² = 17λ² - 140λ + 350. The numerator is (15 - 4λ) + 2(λ - 10) + 2(5) = 5 - 2λ.
3. Square and solve. Setting d = 1/√3 and squaring gives 3(5 - 2λ)² = 17λ² - 140λ + 350. This simplifies to 5λ² - 80λ + 275 = 0, that is λ² - 16λ + 55 = 0, so λ = 5 or λ = 11.
4. So. Both values check out: they give 5/5√3 and 17/17√3. The sum of the possible values of λ is 5 + 11 = 16, which is option (a).

The idea to keep. When the shortest distance is given and a direction has an unknown, square the distance formula to get a quadratic. For the sum of all values, use the sum of roots directly.

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