Question

Cho \(\dfrac{x}{a}=\dfrac{y}{b}=\dfrac{z}{c}\). Rút gọn \(P=\dfrac{x^2+y^2+z^2}{\left(ã+by+cz\right)^2}\)

Answer 1

Đặt: \(\dfrac{x}{a}=\dfrac{y}{b}=\dfrac{z}{c}=k\Rightarrow x=ak;y=bk;z=ck\)

Thay các kết quả trên vào P ta có:

\(P=\dfrac{x^2+y^2+z^2}{\left(ax+by+cz\right)^2}=\dfrac{\left(ak\right)^2+\left(bk\right)^2+\left(ck\right)^2}{\left(a.ak+b.bk+c.ck\right)^2}=\dfrac{a^2k^2+b^2k^2+c^2k^2}{[k\left(a^2+b^2+c^2\right)]^2}=\dfrac{k^2\left(a^2+b^2+c^2\right)}{k^2\left(a^2+b^2+c^2\right)^2}=\dfrac{1}{a^2+b^2+c^2}\)