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\(\left\{{}\begin{matrix}\frac{1}{x}+\frac{1}{y}=2-\frac{1}{z}\\\frac{2}{xy}=4+\frac{1}{z^2}\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}\frac{1}{x^2}+\frac{1}{y^2}+\frac{2}{xy}=4+\frac{1}{z^2}-\frac{4}{z}\\\frac{2}{xy}=4+\frac{1}{z^2}\end{matrix}\right.\)
\(\Rightarrow\frac{1}{x^2}+\frac{1}{y^2}=-\frac{4}{z}\) (1)
Từ pt đầu suy ra:
\(\frac{1}{x}+\frac{1}{y}-2=-\frac{1}{z}\Rightarrow\frac{4}{x}+\frac{4}{y}-8=-\frac{4}{z}\) (2)
Thế (2) vào (1)
\(\frac{1}{x^2}+\frac{1}{y^2}=\frac{4}{x}+\frac{4}{y}-8\)
\(\Leftrightarrow\frac{1}{x^2}-\frac{4}{x}+4+\frac{1}{y^2}-\frac{4}{y}+4=0\)
\(\Leftrightarrow\left(\frac{1}{x}-2\right)^2+\left(\frac{1}{y}-2\right)^2=0\)
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