Question

Cho \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=0;\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=2\)

Tính giá trị \(M=\frac{a^2}{x^2}+\frac{b^2}{y^2}+\frac{z^2}{z^2}\)

Answer 1

\(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=0\Rightarrow\frac{abz+bcx+cay}{abc}=0\)

\(\Rightarrow abz+bcx+cay=0\)

\(\Rightarrow\frac{abz+bcx+cay}{xyz}=0\)

\(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=2\Rightarrow\left(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}\right)^2=4\)

\(\Rightarrow\frac{a^2}{x^2}+\frac{b^2}{y^2}+\frac{c^2}{z^2}+2\left(\frac{ab}{xy}+\frac{bc}{yz}+\frac{ca}{zx}\right)=4\)

\(\Rightarrow M+2\left(\frac{abz+bcx+cay}{xyz}\right)=4\)

\(\Rightarrow M+2.0=4\Rightarrow M=4\)

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