36. 

$$
\left\{\begin{array}{l}
x y=a^{2} \\
\lg ^{2} x+\lg ^{2} y=\frac{5}{2} \lg ^{2} a^{2}
\end{array}\right.
$$
36. We take the logarithm and square both sides of the first equation

$$
\begin{aligned}
& \lg ^{2} x+2 \lg x \lg y+\lg ^{2} y=\lg ^{2} a^{2} \\
& -\lg ^{2} x+\lg ^{2} y=\frac{5}{2} \lg ^{2} a^{2} \\
& 2 \lg x \lg y=-\frac{3}{2} \lg ^{2} a^{2}
\end{aligned}
$$

Consider the system $\left\{\begin{array}{l}\lg x \lg y=-\frac{3}{4} \lg ^{2} a^{2} . \\ \lg x+\lg y=\lg a^{2}\end{array}\right.$

From which $x_{1}=a^{3}, y_{1}=\frac{1}{a}, x_{2}=\frac{1}{a}, y_{2}=a^{3}$.