\(\frac{1}{x^2}+\frac{1}{y^2}=1\Leftrightarrow x^2y^2=x^2+y^2\Leftrightarrow\left(x^2-1\right)\left(y^2-1\right)=1\)
Bình phương 2 vế pt đầu:
\(x^2+y^2-2+2\sqrt{\left(x^2-1\right)\left(y^2-1\right)}=xy+2\)
\(\Leftrightarrow x^2+y^2=xy+2\)
\(\Rightarrow x^2y^2=xy+2\Rightarrow\left[{}\begin{matrix}xy=-1\Rightarrow x^2+y^2=1\\xy=2\Rightarrow x^2+y^2=4\end{matrix}\right.\)
TH1: \(\left\{{}\begin{matrix}x^2+y^2=1\\xy=-1\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}\left(x+y\right)^2-2xy=1\\xy=-1\end{matrix}\right.\) \(\Rightarrow\left(x+y\right)^2=-1\left(vn\right)\)
TH2: \(\left\{{}\begin{matrix}\left(x+y\right)^2-2xy=4\\xy=2\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x+y=2\sqrt{2}\\xy=2\end{matrix}\right.\)
\(\Rightarrow x=y=\sqrt{2}\)
