Question

1.Giải pt :\(x^2+\dfrac{1}{x^2}+y^2+\dfrac{1}{y^2}=4\)

Answer 1

\(x^2+\dfrac{1}{x^2}+y^2+\dfrac{1}{y^2}=4\)

\(\Leftrightarrow x^2-2+\dfrac{1}{x^2}+y^2-2+\dfrac{1}{y^2}=0\)

\(\Leftrightarrow\left(x-\dfrac{1}{x}\right)^2+\left(y-\dfrac{1}{y}\right)^2=0\)

\(\Leftrightarrow\left\{{}\begin{matrix}\left(x-\dfrac{1}{x}\right)^2=0\\\left(y-\dfrac{1}{y}\right)^2=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{1}{x}\\y=\dfrac{1}{y}\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x^2=1\\y^2=1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=\pm1\\y=\pm1\end{matrix}\right.\)