Question

Cho P = \(\frac{x}{\sqrt{xy}+y}+\frac{y}{\sqrt{xy}-x}-\frac{x+y}{\sqrt{xy}}\)

a. Rút gọn P

b. Chứng minh P không đổi khi \(\frac{x}{y}=\frac{x+1}{y+5}\)

Answer 1

ĐKXĐ:...

\(P=\frac{x}{\sqrt{y}\left(\sqrt{x}+\sqrt{y}\right)}+\frac{y}{\sqrt{x}\left(\sqrt{y}-\sqrt{x}\right)}-\frac{x+y}{\sqrt{xy}}\)

\(=\frac{x\sqrt{x}\left(\sqrt{y}-\sqrt{x}\right)+y\sqrt{y}\left(\sqrt{y}+\sqrt{x}\right)-\left(x+y\right)\left(y-x\right)}{\sqrt{xy}\left(y-x\right)}\)

\(=\frac{x\sqrt{xy}-x^2+y^2+y\sqrt{xy}-y^2+x^2}{\sqrt{xy}\left(y-x\right)}\)

\(=\frac{\sqrt{xy}\left(x+y\right)}{\sqrt{xy}\left(y-x\right)}=\frac{y+x}{y-x}\)

\(\frac{x}{y}=\frac{x+1}{y+5}=\frac{x+1-x}{y+5-y}=\frac{1}{5}\Rightarrow y=5x\)

\(\Rightarrow P=\frac{5x+x}{5x-x}=\frac{6x}{4x}=\frac{3}{2}\) (đpcm)