1. Find the ratio $\frac{2023 a c}{b^{2}}$, if it is known that one of the roots of the equation $a x^{2}+b x+c=0$ is 2022 times the other. Here $a \cdot b \cdot c \neq 0$.
Solution. Let $x_{1}=2022 x_{2}$. Then, by Vieta's theorem,

$x_{1}+x_{2}=2023 x_{2}=-\frac{b}{a} \quad x_{1} \cdot x_{2}=2022 x_{2}^{2}=\frac{c}{a}$.

Then

$$
\begin{gathered}
x_{2}=-\frac{b}{2023 a} \\
\frac{c}{a}=\frac{2022 b^{2}}{2023^{2} \cdot a^{2}} \\
a c=\frac{2022 b^{2}}{2023^{2}} \\
2023 a c=\frac{2022 b^{2}}{2023} \\
\frac{2023 a c}{b^{2}}=\frac{2022 b^{2}}{2023 b^{2}}=\frac{2022}{2023}
\end{gathered}
$$

Answer. $\frac{2022}{2023}$