Consider a circle \(C_1: x^2+y^2-4 x-2 y=\alpha-5\). Let its mirror image in the line \(y=2 x+1\) be another circle \(C_…

Consider a circle \(C_1: x^2+y^2-4 x-2 y=\alpha-5\). Let its mirror image in the line \(y=2 x+1\) be another circle \(C_2: 5 x^2+5 y^2-10 f x-10 g y+36=0\). Let \(r\) be the radius of \(C_2\). Then \(\alpha+r\) is equal to ________

[JEE Main 2023, 8 Apr (Shift 1)]