Loc(K)i and trigonometry

Geometry Level pending

The equation of a circle that passes through \((0,1) \) and \((2,3) \) and has its centre on the equation of locus of point which is equidistant from \((-1,1) \) and \((4,-2) \) is \(x^2+y^2 + 2gx + 2fy + c = 0 \).

If \(\tan^{-1} (2g) + \tan^{-1} (2f) + \tan^{-1}(c) = \tan^{-1}(A) \), where \(A\) can be written in the form of \( \dfrac ab\), where \(a\) and \(b\) are coprime positive integers, find \(a+b\).

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