Question

x/a + y/b + z/c = 1 và a/x + b/y + c/z = 0

Tính giá trị của biểu thức A=x^2/a^2 + y^2/b^2 + z^2/c^2

Answer 1

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

\(\left(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\right)^2=\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}+2\left(\frac{xy}{ab}+\frac{yz}{bc}+\frac{xz}{ac}\right)=1\)

\(\Rightarrow\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1-2\left(\frac{abccxy+aabcyz+abbcxz}{aabbcc}\right)=1-2abc\left(\frac{cxy+ayz+bxz}{aabbcc}\right)=1-0=1\)