Denote $M(a,\frac{5}{a})(a>0)$ and $N(\sqrt{6}\cos{\alpha},\sin{\alpha})(0\leq\alpha \leq \frac{\pi}{2})$. We only need prove $MN\geq \frac{9}{7}$. Or , prove $f(a,\alpha)=(\sqrt{6}\cos{\alpha}-a)^{2}+(\frac{5}{a}-\sin{\alpha})^{2}\geq\frac{81}{49}\forall a>0\forall \alpha\in [0,\frac{\pi}{2}]$. Now, i think I can fix $\alpha$, use analysis with function $g(a)=f(a,\alpha)$,...