The shortest distance between the lines (x+7)/-6=(y-6)/7=z and (7-x)/2=y-2=z-6 is :

Solution

Maths sir · 4 steps

Rewrite the second line in standard form first; the 7-x hides a direction component of -2.

1. Read both lines. Line 1 passes through A(-7, 6, 0) with direction b⃗₁ = (-6, 7, 1). For line 2, write (7-x)/2 as (x-7)/-2, so it passes through B(7, 2, 6) with direction b⃗₂ = (-2, 1, 1).

   d = (|AB⃗·(b⃗₁×b⃗₂)|)/(|b⃗₁×b⃗₂|)

2. Cross product. b⃗₁×b⃗₂ = (7-1, -2+6, -6+14) = (6, 4, 8). Its length is √(36+16+64) = √116 = 2√29.
3. Dot product. AB⃗ = (14, -4, 6), so AB⃗·(6, 4, 8) = 84 - 16 + 48 = 116.
4. So. d = 116/2√29 = 58/√29 = 2√29, which is option (a).

The idea to keep. Before reading a direction off a line, make every numerator look like x - x₁. A form like 7-x flips the sign of that component, and the point becomes x₁ = 7.

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