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Yes. The necessary and sufficient condition that the motion is controlled by such a force is one of the equations needed to arrive at the desired equation.Is this type of circular motion actually possible with the single force as shown in the picture ?

No. The trajectory would need to be an ellipse with the source of the force at one focus. Yes, a circle is a special case of an ellipse but its focus would be at the centre, not on the circumference.Is this some sort of planetary motion around the sun with perigee coinciding the focus