A thin wire is bent into a quarter circle of radius, R. The charge on the quarter circle is Q, and nonuniform charge density is lambda(theta)=lambda_0*e^(-theta). Lambda_0 is a constant.
If an electron (or in fact any charge) is placed at the center of the ring (that is origin, i guess)
Then the net force acting on the charge will be exactly zero.
Reasoning 1 : By symmetry the force acting on the charge cannot be in any direction in the plane of the ring.
Then again by symmetry, there is no direction in which the force can be in any direction along the axis.
So there is no direction in which force can be there, therefore there is no force.
Reasoning 2 : Remember that the ring is a circle, if charge at some point A on the circle applies a force F at the center, then there is a point opposite to B on the circle which applies a force -F at the center. Hence cancelling the force.