Let λ be an integer. If the shortest distance between the lines
x - λ = 2y - 1 = -2z and x = y + 2λ = z - λ is √7/2√2, then the value of | λ | is _________.

Solution

Maths sir · 4 steps

Rewrite both lines so x, y and z each have coefficient 1, then use the cross-product distance formula.

1. Standard form. Line 1: x-λ = 2(y-1/2) = -2z. Divide through by 2 to get (x-λ)/2 = (y-1/2)/1 = z/-1, through A(λ, 1/2, 0) with direction b₁⃗ = (2, 1, -1). Line 2: x/1 = (y+2λ)/1 = (z-λ)/1, through B(0, -2λ, λ) with direction b₂⃗ = (1, 1, 1).
2. Common perpendicular. b₁⃗ × b₂⃗ = (1+1, -1-2, 2-1) = (2, -3, 1), with magnitude √14. Also AB⃗ = (-λ, -2λ-1/2, λ).

   d = (|AB⃗·(b₁⃗×b₂⃗)|)/(|b₁⃗×b₂⃗|) = (|-2λ + 6λ + 3/2 + λ|)/√14 = (|5λ + 3/2|)/√14

3. Equate. Set this equal to √7/2√2. Since √7/2√2 × √14 = 7√2/2√2 = 7/2, we need |5λ + 3/2| = 7/2. This gives λ = 2/5 or λ = -1.
4. So. λ must be an integer, so λ = -1 and |λ| = 1.

The common slip: 0.4. Keeping only the root of 5λ + 3/2 = 7/2 gives λ = 0.4, which is not an integer. The modulus has a second root, λ = -1.

The idea to keep. Before reading off a direction, make the coefficient of each of x, y and z equal to 1. Then the shortest distance is the projection of the joining vector onto b₁⃗×b₂⃗.

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