Let x ^ 2 + y ^ 2 + Ax + By + C = 0 be a circle passing through (0,...

Equation of tangent at $\left(2, 4\right)$ to the parabola $y = {x}^{2}$ is $\dfrac{1}{2}\left(y + 4\right) = 2x$

$⇒ 4x - y - 4 = 0$

Then family of circle touching this tangent at \left(2, 4\right)$is</p><p>${\left(x - 2\right)}^{2} + {\left(y - 4\right)}^{2} + \lambda\left(4x - y - 4\right)$= 0 . . . \left(i\right)$

Since it passes through \left(0, 6\right)$, we get,</p><p>$4 + 4 + \lambda\left(- 10\right) = 0 ⇒ \lambda = \dfrac{4}{5}$</p><p>So, equation \left(i\right)$ becomes ${x}^{2} + {y}^{2} - \dfrac{4}{5}x - \dfrac{44}{5}y + \dfrac{84}{5} = 0$

Hence, $A + C = \dfrac{80}{5} = 16$