Conic Section

\[ \begin{aligned} & x^{2}+y^{2}+A x+B y+C=0 \\ & \Rightarrow 6 B+C=-36 \end{aligned} \]

The tangent of the parabola \( y=x^{2} \) at \( (2,4) \) is

\[ 4 x-y-4=0 \]

The tangent of circle \( x^{2}+y^{2}+A x+B y+C=0 \) at

\( (2,4) \) is

\[ (4+A) x+(8+B) y+2 A+4 B+2 C=0 \]

From equation (1) and (2)

\[ \begin{aligned} & \frac{4+A}{4}=\frac{8+B}{-1}=\frac{2 A+4 B+2 C}{-4} \\ & A+4 B=-36 \\ & 3 A+4 B+2 C=-4 \end{aligned} \]

From equation (3) and (4)

\[ A+C=16 \]