- a) căn(x-1/2y+1) + căn(2y+1/x-1) = 2 và x-y=1
ĐKXĐ: ..........
[tex]\left\{\begin{matrix} \sqrt{\frac{x-1}{2y+1}}+\sqrt{\frac{2y+1}{x-1}}=2(1) & \\ x-y=1(2) & \end{matrix}\right.[/tex]
Theo BĐT AM-GM, từ (1) ta có:
[tex]VT=\sqrt{\frac{x-1}{2y+1}}+\sqrt{\frac{2y+1}{x-1}}\geq 2\sqrt{\sqrt{\frac{x-1}{2y+1}.\frac{2y+1}{x-1}}}=2=VP\\ \Rightarrow \sqrt{\frac{x-1}{2y+1}}=\sqrt{\frac{2y+1}{x-1}}\Leftrightarrow \frac{x-1}{2y+1}=\frac{2y+1}{x-1}=1[/tex]
Mặt khác, từ (2) ta có:
[tex]\frac{x-1}{2y+1}=\frac{x-1}{2(x-1)+1}=1\Leftrightarrow x=0\Leftrightarrow y=-1[/tex] (đối chiếu ĐKXĐ nha)