Question

Cho 3 số x,y vả z thoả mãn 1/x+1/y+1/z=0. Hãy tính A= yz/x^2+zx/y^2+xy/z^2

Answer 1

CÓ:\(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0\)

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

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

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

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

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