The centre of the circle passing through the

$y=x^{2}$

$\left.\frac{\mathrm{dy}}{\mathrm{dx}}\right|_{\mathrm{P}}=4$

$(y-4)=4(x-2)$

$4 x-y-4=0$

Circle : $(x-2)^{2}+(y-4)^{2}+\lambda(4 x-y-4)=0$

passes through $(0,1)$

$4+9+\lambda(-5)=0 \Rightarrow \lambda=\frac{13}{5}$

Circle $: x^{2}+y^{2}+x(4 \lambda-4)+y(-\lambda-8)+(20-4 \lambda)=0$

Centre : $\left(2-2 \lambda, \frac{\lambda+8}{2}\right) \equiv\left(\frac{-16}{5}, \frac{53}{10}\right)$