Right answer is (a) When they are equal distance from the junction

Explanation: Let OX and OY be two straight RAILWAY lines and they meet at O at right angles.

One train starts from the junction and moves with uniform velocity u km/hr along the line OY.

And at the same INSTANT, another train starts towards the junction O from station A on the line OX with same uniform velocity u km/hr.

Let C and B be the position of the two trains on lines OY and OX respectively after t hours from the start.

Then OC = AB = ut km. Join BC and let OA = a km and BC = x km.

Then OB = a – ut.

Now, from the right angled triangle BOC we get,

BC^2 = OB^2 + OC^2

Or x^2 = (a – ut)^2 + (ut)^2

Thus, d(x^2)/dt = 2(a – ut)(-u) + u^2(2t)

And d^2(x^2)/dt^2 = 2u^2 + 2u^2 = 4u^2

For MAXIMUM or minimum value of x^2(i.e., x) we must have,

d(x^2)/dt = 0

Or 2(a – ut)(-u) + u^2(2t) = 0

Or 2ut = a[Since u ≠ 0]

Or t = a/2u

Again at t = a/2u we have, d^2(x^2)/dt^2 = 4u^2 > 0

Therefore, x^2(i.e., x) is minimum at t = a/2u

Now when t= a/2u, then OC = ut = u(a/2u) = a/2 and OB =a – ut = a – u(a/2u) = a/2 that is at t = a/2u we have, OC = OB.

Therefore, the trains are nearest to each other when they are equally distant from the junction.