\[\boxed{\text{447\ (447).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]
\[\textbf{а)}\ \left\{ \begin{matrix} x^{2} + y^{2} = 12 \\ xy = - 6\ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} x = - \frac{6}{y}\text{\ \ \ \ \ \ \ \ \ \ \ } \\ \frac{36}{y^{2}} + y^{2} = 12 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} x = - \frac{6}{y}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ 36 + y^{4} - 12y^{2} = 0 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} x = - \frac{6}{y}\text{\ \ \ \ \ \ \ \ \ \ \ \ } \\ \left( y^{2} - 6 \right)^{2} = 0 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} y_{1} = \sqrt{6}\text{\ \ \ \ } \\ x_{1} = - \sqrt{6} \\ \end{matrix} \right.\ \text{\ \ \ \ }или\ \ \ \]
\[\left\{ \begin{matrix} y_{2} = - \sqrt{6} \\ x_{2} = \sqrt{6.}\text{\ \ \ } \\ \end{matrix} \right.\ \]
\[\textbf{б)}\ \left\{ \begin{matrix} 2x^{2} - y^{2} = 34 \\ xy = 20\ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} x = \frac{20}{y}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ 2 \cdot \frac{400}{y^{2}} - y^{2} = 34 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} x = \frac{20}{y}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ 800 - y^{4} - 34y^{2} = 0 \\ \end{matrix} \right.\ \Longrightarrow\]
\[\Longrightarrow \left\{ \begin{matrix} x = \frac{20}{y}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ y^{4} + 34y^{2} - 800 = 0 \\ \end{matrix} \right.\ \]
\[Пусть\ t = y^{2};\ \ t \geq 0:\]
\[t^{2} + 34t - 800 = 0\]
\[D = 17^{2} + 800 = 1089\]
\[t_{1,2} = - 17 \pm 33\]
\[так\ как\ t \geq 0:\]
\[t = 16 \Longrightarrow y^{2} = 16.\]
\[1)\ y_{1} = 4;\ \ x_{1} = 5;\]
\[2)y_{2} = - 4;\ \ x_{2} = - 5.\]
\[Ответ:а)\ \left( - \sqrt{6};\sqrt{6} \right);\ \ \left( \sqrt{6};\ - \sqrt{6} \right);\]
\[\ \ \ \ \ \ \ \ \ \ \ \ \ \ б)\ (5;4);\ \ ( - 5;\ - 4).\]