$\begin{array}{l}
\dfrac{1}{{O{G^2}}} = \dfrac{1}{{S{O^2}}} + \dfrac{1}{{O{F^2}}}\\
\Rightarrow \dfrac{1}{{O{G^2}}} = \dfrac{1}{{\dfrac{{3{a^2}}}{4}}} + \dfrac{1}{{\dfrac{{3{a^2}}}{{16}}}}\\
\Rightarrow \dfrac{1}{{O{G^2}}} = \dfrac{4}{{3{a^2}}} + \dfrac{{16}}{{3{a^2}}} = \dfrac{{20}}{{3{a^2}}}\\
\Rightarrow O{G^2} = \dfrac{{3{a^2}}}{{20}} \Rightarrow OG = \dfrac{{a\sqrt 3 }}{{\sqrt {20} }} = \dfrac{{a\sqrt {60} }}{{20}}
\end{array}$