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

Đề bài: Cho $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$  và $\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=0$ . Chứng minh rằng: $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ .

Trả lời

Hướng dẫn giải:

Ta có:

+) $$\begin{gathered} \frac{a}{x}+\frac{b}{y}+\frac{c}{z}=0\iff \frac{ayz+bxz+cxy}{xyz}=0 \\ \iff ayz+bxz+cxy=0 \end{gathered}$$

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

$$\begin{gathered} \iff \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}+2\left(\frac{xy}{ab}+\frac{yz}{bc}+\frac{xz}{zc}\right)=1 \\ \iff \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}+2\left(\frac{ayz+bxz+cxy}{abc}\right)=1 \end{gathered}$$

$\iff \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ (đpcm).