Question

cho \(\frac{xy}{x+y}=\frac{yz}{y+z}\)\(=\frac{xz}{x+z}\)

Tính \(M=\frac{x^2+y^2+z^2}{xy+yz+xz}\)

 

Answer 1

Ta có:\(\frac{xy}{x+y}=\frac{yz}{y+z}\Rightarrow xy\left(y+z\right)=yz\left(x+y\right)\Leftrightarrow xy^2+xyz=xyz+y^2z\Leftrightarrow xy^2=y^2z\Rightarrow x=z\)(1)

\(\frac{yz}{y+z}=\frac{xz}{x+z}\Rightarrow yz\left(x+z\right)=xz\left(y+z\right)\Leftrightarrow xyz+yz^2=xyz+xz^2\Leftrightarrow yz^2=xz^2\Rightarrow y=x\)(2)

Từ (1)và(2)suy ra:x=y=z

\(\Rightarrow x^2=xy,y^2=yz,z^2=xz\)

\(\Rightarrow M=\frac{xy+yz+xz}{xy+yz+xz}=1\)

Vậy M=1

Answer 2

\(x^2=xy,y^2=yz,z^2=xz\)

là sao??