If the foot of the perpendicular from point (4, 3, 8) on the line L₁:(x - a)/l = (y - 2)/3 = (z - b)/4, l ≠ 0 is (3, 5, 7), then the shortest distance between the line L₁ and line L₂:(x - 2)/3 = (y - 4)/4 = (z - 5)/5 is equal to :

Solution

Maths sir · 5 steps

The foot lies on the line and the perpendicular meets it at right angles. Those two facts give a, b and l.

1. Foot lies on L1. Put (3, 5, 7) into L₁. The y-part gives (5-2)/3 = 1, so (3-a)/l = 1 and (7-b)/4 = 1. That gives b = 3 and a = 3 - l.
2. Right angle condition. The segment from the foot to the point is (4-3, 3-5, 8-7) = (1, -2, 1). It is perpendicular to the direction (l, 3, 4), so l - 6 + 4 = 0. That gives l = 2 and a = 1.
3. Set up both lines. L₁ passes through (1, 2, 3) with direction b⃗₁ = (2, 3, 4). L₂ passes through (2, 4, 5) with direction b⃗₂ = (3, 4, 5). Joining the two points gives a⃗₂ - a⃗₁ = (1, 2, 2).

   d = (|(a⃗₂ - a⃗₁)·(b⃗₁ × b⃗₂)|)/(|b⃗₁ × b⃗₂|)

4. Compute. b⃗₁ × b⃗₂ = (15-16, 12-10, 8-9) = (-1, 2, -1), with magnitude √6. The dot product is (1)(-1) + (2)(2) + (2)(-1) = 1.
5. So. The shortest distance is d = 1/√6, which is option (a).

The idea to keep. When a foot of perpendicular hides unknowns in a line, use two facts: the foot satisfies the line, and the perpendicular dotted with the direction is zero.

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