Consider a two body system say the earth with mass $M$ and the moon with mass $m$ and distance $d$ between them. Thus there is a point in space between the earth and the moon where the force of gravity on an arbitrary object is equal in magnitude but opposite in direction. Similarly on the other side of the moon, there will be a point in space where the force of gravity on an arbitrary object is equal in magnitude but and identical in direction.
What is the locus of all points where the magnitude of the gravitational force on an arbitrary object due to the earth and the moon is equal in magnitude regardless of the direction. Will it be an ellipse with the moon at one focus?
What will be the path of motion of an object if it suddenly pops into existence with zero initial velocity on the above locus? Will it revolve around the larger body or will it revolve around the smaller body or will it revolve around their common center of mass or will it float aimlessly in space?
How will the above answers change if we consider the two body system to be an isolated system with no external gravitational influence