\[\boxed{\text{571}\text{\ (571)}\text{.}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]
\[\left\{ \begin{matrix} x^{2} + y^{2} = 45 \\ y = 2x\ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} x^{2} + 4x^{2} = 45 \\ y = 2x\ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \Longrightarrow \left\{ \begin{matrix} x^{2} = 9\ \\ y = 2x \\ \end{matrix} \right.\ \]
\[1)\ \left\{ \begin{matrix} x_{1} = 3 \\ y_{1} = 6 \\ \end{matrix} \right.\ \text{\ \ \ \ }или\ \ \ \left\{ \begin{matrix} x_{2} = - 3 \\ y_{2} = - 6 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow не\ удовлетворяют\ \]
\[условию\ x > 0,\ y > 0.\]
\[Ответ:x = 3,\ y = 6.\]