Задача 22. Решите систему уравнений: \(\left\{ \begin{array}{l}2{x^2} + {y^2} = 36,\\8{x^2} + 4{y^2} = 36x.\end{array} \right.\)
Ответ
ОТВЕТ: \(\left( {4;\,-2} \right),\,\,\,\;\left( {4;\,2} \right).\)
Решение
Разделим обе части второго уравнения на 4. Тогда система уравнений примет вид:
\(\left\{ \begin{array}{l}2{x^2} + {y^2} = 36,\\2{x^2} + {y^2} = 9x\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}9x = 36,\\2{x^2} + {y^2} = 36\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}x = 4,\\2{x^2} + {y^2} = 36\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}x = 4,\\2 \cdot {4^2} + {y^2} = 36\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}x = 4,\\{y^2} = 4\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}x = 4,\\\left[ \begin{array}{l}y = -2,\\y = 2\end{array} \right.\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left[ \begin{array}{l}\left\{ \begin{array}{l}x = 4,\\y = -2,\end{array} \right.\\\left\{ \begin{array}{l}x = 4,\\y = 2.\end{array} \right.\end{array} \right.\)
Ответ: \(\left( {4;\,-2} \right),\,\,\,\;\left( {4;\,2} \right).\)