The center of the circle passing through the point $\left(0,1\right)$ and touching the parabola $y=x^2$  at the point $\left(2,4\right)$ is $\left(\mathrm{\alpha },\mathrm{\beta }\right)$ then $\mathrm{\alpha }+2\mathrm{\beta }=$____
 

Equation of the tangent at (2,4) of the parabola ${\mathrm{x}}^2=\mathrm{y}$ is 
$\mathrm{x}\left(2\right)=\frac{1}{2}(\mathrm{y}+4)\Rightarrow 4\mathrm{x}-\mathrm{y}-4=0$
Required circle be $\mathrm{S}+\mathrm{\lambda L}=0$
$\Rightarrow {(x-2)}^2+{(y-4)}^2+\lambda (4x-y-4)=0$
If passes through the point $\left(0,1\right)$.
$\begin{array}{l}\therefore 4+9+\mathrm{\lambda }(-5)=0\Rightarrow \mathrm{\lambda }=\frac{13}{5}\\ \therefore {\mathrm{x}}^2+{\mathrm{y}}^2-4\mathrm{x}+{\mathrm{y}}^2-8\mathrm{y}+20+\frac{13}{5}(4\mathrm{x}-\mathrm{y}-4)=0\\ \mathrm{C}\left(\frac{-16}{5},\frac{53}{10}\right).\end{array}$