The correct answer is option 3 - a parabola.

In physics, the displacement-time graph represents the motion of an object in one dimension. When a particle is acted upon by a constant force, it undergoes a uniformly accelerated motion. This means that its displacement changes at a constant rate over time.

A parabolic shape represents uniformly accelerated motion on a displacement-time graph. When a particle undergoes uniformly accelerated motion, its displacement changes as a function of time squared. This is because the rate of change of velocity, or acceleration, is constant.

A straight line (option 1) would represent uniform motion, where the particle moves with a constant velocity. A circle (option 2) represents periodic motion or circular motion, which is not applicable in the case of a particle acted upon by a constant force.

Option 4, which states that any curve is possible depending upon initial conditions, is incorrect. The displacement-time graph of a particle under constant force will always follow a parabolic shape.

Overall, the correct answer is option 3 - a parabola.