Question

[nguyen kim chi]

cho 3 so huu ti a;b;c thoa man 1/a+1/b=1/c

c/m  \(\sqrt{a^2+b^2+c^2}\) la so huu ti

Answer 1

[Mr Lazy]

\(\frac{1}{a}+\frac{1}{b}=\frac{1}{c}\Rightarrow c=\frac{ab}{a+b}\)

\(a^2+b^2+c^2=\left(a+b\right)^2-2ab+\frac{a^2b^2}{\left(a+b\right)^2}=\frac{\left(a+b\right)^4-2ab\left(a+b\right)^2+a^2b^2}{\left(a+b\right)^2}\)

\(=\frac{\left[\left(a+b\right)^2-ab\right]^2}{\left(a+b\right)^2}\)

\(\Rightarrow\sqrt{a^2+b^2+c^2}=\left|\frac{\left(a+b\right)^2-ab}{a+b}\right|\) là số hữu tỉ.