Let l₁ be the line in xy-plane with x and y intercepts 1/8 and 1/4√2 respectively, and l₂ be the line in zx-plane with x and z intercepts - 1/8 and - 1/6√3 respectively. If d is the shortest distance between the line l₁ and l₂, then d⁻² is equal to _______________.

Solution

Maths sir · 4 steps

Read each line off its two intercepts, cross the directions, then project the gap between the points.

1. Line l1. l₁ lies in the xy-plane, so it passes through A(1/8, 0, 0) and (0, 1/4√2, 0). The direction is (-1/8, 1/4√2, 0). Multiply by -8 to get b⃗ = (1, -√2, 0).
2. Line l2. l₂ lies in the zx-plane, so it passes through C(-1/8, 0, 0) and (0, 0, -1/6√3). The direction is (1/8, 0, -1/6√3). Multiply by 24√3 to get d⃗ = (3√3, 0, -4).

   d = (|AC⃗ · (b⃗ × d⃗)|)/(|b⃗ × d⃗|)

3. Cross and dot. b⃗ × d⃗ = (4√2, 4, 3√6), so |b⃗ × d⃗|² = 32 + 16 + 54 = 102. Also AC⃗ = (-1/4, 0, 0), so the dot product is -1/4 × 4√2 = -√2.
4. So. d = √2/√102 = 1/√51. This gives d² = 1/51, so d⁻² = 51.

The common slip: 204. Taking the gap between A and C as 1/8 instead of 1/8 + 1/8 = 1/4 halves d and gives 204, not 51.

The idea to keep. A line in a coordinate plane is fixed by its two intercepts, which give a point and a direction. For skew lines, project the vector joining one point on each line onto b⃗ × d⃗.

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