ĐKXĐ: \(a>0;a\ne4\)
\(P=\left(\frac{\sqrt{a}-2}{\left(\sqrt{a}+2\right)\left(\sqrt{a}-2\right)}+\frac{\sqrt{a}+2}{\left(\sqrt{a}-2\right)\left(\sqrt{a}+2\right)}\right).\frac{\sqrt{a}-2}{\sqrt{a}}\)
\(=\left(\frac{2\sqrt{a}}{\left(\sqrt{a}+2\right)\left(\sqrt{a}-2\right)}\right)\left(\frac{\sqrt{a}-2}{\sqrt{a}}\right)=\frac{2}{\sqrt{a}+2}\)
Để \(P>\frac{1}{3}\Leftrightarrow\frac{2}{\sqrt{a}+2}>\frac{1}{3}\Rightarrow\sqrt{a}+2< 6\Rightarrow a< 16\) \(\Rightarrow\left\{{}\begin{matrix}0< a< 16\\a\ne4\end{matrix}\right.\)
\(Q=\frac{9}{2}P=\frac{9}{\sqrt{a}+2}\)
Để Q nguyên \(\Rightarrow\sqrt{a}+2=Ư\left(9\right)\)
Mà \(\sqrt{a}+2>2\Rightarrow\sqrt{a}+2=\left\{3;9\right\}\)
\(\Rightarrow\sqrt{a}=\left\{1;7\right\}\Rightarrow a=\left\{1;49\right\}\)