Question

[quanhong123]

cho x/a +y/b+z/c=1 va a/x+b/y+c/z=0 tinh xx/aa+yy/bb+zz/cc

Answer 1

[Đinh quang hiệp]

\(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=\frac{ayz+bxz+cxy}{xyz}=0\Rightarrow ayz+bxz+cxy=0\)

\(\left(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\right)^2=\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}+2\left(\frac{xy}{ab}+\frac{xz}{ac}+\frac{yz}{bc}\right)\)

\(=\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}+2\left(\frac{cxy+bxz+ayz}{abc}\right)=\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}+2\cdot\frac{0}{abc}=\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}\)

\(\Rightarrow\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=\left(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\right)^2\)mà \(\left(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\right)^2=1^2=1\)

\(\Rightarrow\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\)