18.218 ** Given the line $y=k x+b(b \neq 0)$ intersects the quadratic curve
$$
A x^{2}+2 B x y+C y^{2}+2 D x+2 E y+F=0
$$

at points $M$ and $N$. Find the necessary and sufficient condition for the lines $O M$ and $O N$ to be perpendicular.
Let the coordinates of $M$ and $N$ be $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)$. Substituting the equation of the line into the equation of the conic section and simplifying, we get
$$
\left(A+2 B k+C k^{2}\right) x^{2}+(2 B b+2 E k+2 C k b+2 D) x+2 E b+C b^{2}+F=0 .
$$

By Vieta's formulas,
$$
x_{1}+x_{2}=-\frac{2 B b+2 E k+2 C k b+2 D}{A+2 B k+C k^{2}}, x_{1} x_{2}=\frac{2 E b+C b^{2}+F}{A+2 B k+C k^{2}} .
$$

Also, $y_{1}=k x_{1}+b, y_{2}=k x_{2}+b$, so $y_{1} y_{2}=k^{2} x_{1} x_{2}+\left(x_{1}+x_{2}\right) k b+b^{2}$. Therefore,
$$
\begin{aligned}
O M \perp O N & \Leftrightarrow y_{1} y_{2}+x_{1} x_{2}=0 \Leftrightarrow\left(k^{2}+1\right) x_{1} x_{2}+\left(x_{1}+x_{2}\right) k b+b^{2}=0 \\
& \Leftrightarrow A b^{2}-2 D k b+F k^{2}+C b^{2}+2 E b+F=0,
\end{aligned}
$$

which means the necessary and sufficient condition for the lines $O M$ and $O N$ to be perpendicular is that equation (1) holds.