Consider 2D motion of a particle,
For a particle to move in a circle there must be a force of constant magnitude acting on it always pointing towards the centre of the circle.
Similarly for an ellipse there must be a force acting on it with a magnitude inversely proportional to the square of the distance from one of its focus and always pointing towards that focus.
For a parabolic path there must be a force with a constant magnitude and direction acting on it.
However I do not know the conditions for a hyperbola and a rectangular hyperbola. Any help will be greatly appreciated. Also correct me if I'm wrong about any of my assumptions.
Edit: I tried to find the required acceleration of the particle by differentiating $x=a\sec(ωt)$ and $y=b\tan(ωt)$
I got the acceleration as $ax=(ω^2)x(2((x^2)/(a^2)) - 1)$ and $ay=2(ω^2)y(((y^2)/(b^2)) + 1)$
This seems to suggest that the force acting on the particle increases with its distance from focus. Is this correct?