Question

Cho a,b > 0, c ≠ 0. CMR:

\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0\Leftrightarrow\sqrt{a+b}=\sqrt{a+c}+\sqrt{b+c}\)

Answer 1

Lời giải:

Ta có:

\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0\)

\(\Leftrightarrow ab+bc+ac=0(*)\).

Từ $(*)$ ta thấy: \(c=\frac{-ab}{a+b}< 0\) do $a,b>0$

\(c+a=\frac{-ac}{b}>0\) do $c< 0; a,b>0$

\(c+b=\frac{-bc}{a}>0\) do $c< 0; a,b>0$

Do đó:

\((*)\Leftrightarrow c^2+ab+bc+ac=c^2\)

\(\Leftrightarrow (c+a)(c+b)=c^2\)

\(\Leftrightarrow \sqrt{(c+a)(c+b)}=|c|=-c\)

\(\Leftrightarrow 2\sqrt{(c+a)(c+b)}+2c=0\)

\(\Leftrightarrow (c+a)+(c+b)+2\sqrt{(c+a)(c+b)}=a+b\)

\(\Leftrightarrow (\sqrt{c+a}+\sqrt{c+b})^2=a+b\)

\(\Leftrightarrow \sqrt{c+a}+\sqrt{c+b}=\sqrt{a+b}\) (đpcm)