Object moving in the xy-plane is acted on by a conservative force described by the potential-energy function u (x, y) = α (1/x2+1/y2), where α is a positive constant

It seems like the problem is asking for us to derive an expression for the force F⃗ stated in terms of the unit vectors i^ and j^.

The solution for this is:

U (x, y) = α (x^-2 + y^-2)

F = - dU (x, y)

F = - (dU/dx + dU/dy)

F = - (-2αx^-3 - 2αy^-3)

F = 2α (x^-3 + y^-3)

F = 2α (1/x^3 i + 1/y^3 j)