Задача 9. Решите систему уравнений \(\left\{ {\,\begin{array}{*{20}{c}} {{x^2}-y = 0\,\,\,\,\,} \\ {{x^2} + {y^2} = 5y} \end{array}} \right.\)
Ответ
ОТВЕТ: \(\left( {\,0;\,0} \right),\;\left( {\,2;\,4} \right),\) \(\left( {\,-2;\,4} \right)\).
Решение
\(\left\{ {\begin{array}{*{20}{c}}{{x^2}-y = 0,\,\,\,\,}\\{{x^2} + {y^2} = 5y}\end{array}} \right.\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{y = {x^2},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{{x^2} + {x^4} = 5{x^2}}\end{array}} \right.\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{y = {x^2},\,\,\,\,\,\,\,\,\,\,\,\,\,}\\{{x^4}-4{x^2} = 0}\end{array}\,\,\,\,\,\,\, \Leftrightarrow } \right.\)
\( \Leftrightarrow \,\,\,\,\,\,\,\left\{ {\begin{array}{*{20}{c}}{y = {x^2},}\\{\left[ {\begin{array}{*{20}{c}}{x = 0,\,\,}\\{x = 2,\,}\\{x = -2}\end{array}} \right.}\end{array}} \right.\,\,\,\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\left\{ {\begin{array}{*{20}{c}}{x = 0,}\\{y = 0,}\end{array}\,\,} \right.}\\{\left\{ {\begin{array}{*{20}{c}}{x = 2,}\\{y = 4,}\end{array}\,\,} \right.}\\{\left\{ {\begin{array}{*{20}{c}}{x = -2,}\\{y = 4.\,\,}\end{array}} \right.}\end{array}} \right.\)
Ответ: \(\left( {0;0} \right),\,\,\,\left( {2;4} \right),\,\,\,\left( {-2;4} \right).\)