Задача 21. Решите систему уравнений: \(\left\{ \begin{array}{l}5{x^2} + {y^2} = 36,\\10{x^2} + 2{y^2} = 36x.\end{array} \right.\)
Ответ
ОТВЕТ: \(\left( {2;\,-4} \right),\,\,\,\;\left( {2;\,4} \right).\)
Решение
Разделим обе части второго уравнения на 2. Тогда система уравнений примет вид:
\(\left\{ \begin{array}{l}5{x^2} + {y^2} = 36,\\5{x^2} + {y^2} = 18x\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}18x = 36,\\5{x^2} + {y^2} = 36\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}x = 2,\\5{x^2} + {y^2} = 36\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \)
\( \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}x = 2,\\5 \cdot {2^2} + {y^2} = 36\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}x = 2,\\{y^2} = 16\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \begin{array}{l}x = 2,\\\left[ \begin{array}{l}y = -4,\\y = 4\end{array} \right.\end{array} \right.\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left[ \begin{array}{l}\left\{ \begin{array}{l}x = 2,\\y = -4,\end{array} \right.\\\left\{ \begin{array}{l}x = 2,\\y = 4.\end{array} \right.\end{array} \right.\)
Ответ: \(\left( {2;\,-4} \right),\,\,\,\;\left( {2;\,4} \right).\)