Question

Cho x>0 , y>0 và \(\frac{1}{\text{x}}+\frac{1}{y}\)= 1

C/m : \(\sqrt{\text{x+y}}=\sqrt{\text{x}-1}+\sqrt{y-1}\)

Answer 1

\(\frac{1}{x}+\frac{1}{y}=1\Leftrightarrow\frac{x+y}{xy}=1\Rightarrow x+y=xy\Rightarrow\sqrt{x+y}=\sqrt{xy}\)

\(\frac{1}{x}=1-\frac{1}{y}=\frac{y-1}{y}\Rightarrow y-1=\frac{y}{x}\Rightarrow\sqrt{y-1}=\sqrt{\frac{y}{x}}\)

Tương tự ta có \(\sqrt{x-1}=\sqrt{\frac{x}{y}}\)

\(\Rightarrow\sqrt{x-1}+\sqrt{y-1}=\sqrt{\frac{x}{y}}+\sqrt{\frac{y}{x}}=\frac{x+y}{\sqrt{xy}}=\frac{x+y}{\sqrt{x+y}}=\sqrt{x+y}\)