a) Yes b) No
Work Step by Step
a) It would be possible if the force was 45 degrees below the negative x-axis with magnitude $\sqrt2F$, for this would cancel out both of the other forces and thus cause the net force to be zero. We place the force at the point (0,1). After all, since we can pick the axis of rotation, we can arbitrarily call this point the axis of rotation. When this is done, we see that the net torque is 0, for two forces are on the axis of rotation, and the third force is parallel to the axis of rotation and thus causes no torque. b) The object already has 0 net force on it, so it impossible to add a force and still keep 0 net force, meaning that a force cannot be added while static equilibrium is maintained.