The centre of the circle passing through the point $(0,1)$ and touching the parabola $y=x^{2}$ at the point $(2,4)$ is\\
(1) $\left(\frac{-53}{10},\frac{16}{5}\right)$\\
(2) $\left(\frac{6}{5},\frac{53}{10}\right)$\\
(3) $\left(\frac{3}{10},\frac{16}{5}\right)$\\
(4) $\left(\frac{-16}{5},\frac{53}{10}\right)$