Question

cho x,y,z khác 0 thỏa mãn x+y+z=xyz và 1/x+1/y+1/z=\(\sqrt{3}\)

tính giá trị của 1/x²+1/y²+1/z²

Answer 1

\(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\sqrt{3}\)

\(\Leftrightarrow\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)^2=3\)

\(\Leftrightarrow\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}+2\left(\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx}\right)=3\)

\(\Leftrightarrow\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}+\frac{2\left(x+y+z\right)}{xyz}=3\)

\(\Leftrightarrow\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}+\frac{2xyz}{xyz}=3\)

\(\Leftrightarrow\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}+2=3\)

\(\Leftrightarrow\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=1\)