If the shortest distance between the line joining the points (1, 2, 3) and (2, 3, 4), and the line (x - 1)/2 = (y + 1)/(- 1) = (z - 2)/0 is α, then 28α² is equal to ____________.

Solution

Maths sir · 3 steps

Get a point and a direction for each line, then project the joining vector onto b⃗₁ × b⃗₂.

1. Read off both lines. Line 1 passes through A(1, 2, 3) with direction b⃗₁ = (2-1, 3-2, 4-3) = (1, 1, 1). Line 2 passes through B(1, -1, 2) with direction b⃗₂ = (2, -1, 0).
2. Common perpendicular. b⃗₁ × b⃗₂ = (1 · 0 - 1 · (-1), 1 · 2 - 1 · 0, 1 · (-1) - 1 · 2) = (1, 2, -3). Its magnitude is √(1 + 4 + 9) = √14.

   α = (|AB⃗ · (b⃗₁ × b⃗₂)|)/(|b⃗₁ × b⃗₂|) = (|(0, -3, -1) · (1, 2, -3)|)/√14 = (|0 - 6 + 3|)/√14 = 3/√14

3. So. α² = 9/14, so 28α² = 28 × 9/14 = 18.

The common slip: 162. Forgetting the minus sign on the middle component of the cross product gives (1, -2, -3), so α = 9/√14 and 28α² = 162, not 18.

The idea to keep. For skew lines, the shortest distance is the joining vector projected onto b⃗₁ × b⃗₂. Expand the cross product carefully, because the middle component carries a minus sign.

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