Let x² + y² + Ax + By + C = 0 be a circle passing through (0, 6) and touching the parabola y = x² at (2, 4). Then A + C is equal to ___________.

For tangent to parabola $y=x^{2}$ at $(2,4)$ <br/><br/> $\left.\frac{d y}{d x}\right|_{(2,4)}=4$ <br/><br/> Equation of tangent is $y-4=4(x-2)$ <br/><br/> $\Rightarrow 4 x-y-4=0$ <br/><br/> Family of circle can be given by <br/><br/> $(x-2)^{2}+(y-4)^{2}+\lambda(4 x-y-4)=0$ <br/><br/> As it passes through $(0,6)$ <br/><br/> $2^{2}+2^{2}+\lambda(-10)=0$ <br/><br/> $\Rightarrow \lambda=\frac{4}{5}$ <br/><br/> Equation of circle is <br/><br/> $$ \begin{aligned} &(x-2)^{2}+(y-4)^{2}+\frac{4}{5}(4 x-y-4)=0 \\\\ &\Rightarrow \left(x^{2}+y^{2}-4 x-8 y+20\right)+\left(\frac{16}{5} x-\frac{4}{5} y-\frac{16}{5}\right)=0 \\\\ &A=-4+\frac{16}{5}, C=20-\frac{16}{5} \end{aligned} $$ <br/><br/> So, $A+C=16$