The equation of circle passing through
intersection points of two circles is given by S₁ + AS₂ = 0 (where S₁ and S2 are the
equations of given circle)
The centre of circle passing through
(2, 3), (4,1) and (0, 0) is
Not sure what the " S₁ + AS₂ = 0" stuff is all about, since in general two circles need not intersect, and there are infinitely many circles that pass through the point(s) if they do, and I don't even know what it means to try and take a linear combination of two equations to form a third equation.
As for the three given points, you can either use them to get three equations in h,k,r where (x-h)^2 + (y-k)^2 = r^2
or if the points are A,B,C, then find the perpendicular bisectors of the chords AB and AC, and the center O of the circle is where they intersect. Then you can find the distance r = OA
Post your work if you get stuck.