I found the answer I was looking for.
The thing to use here is the construct of
\left \middle \right
For example, in this case, two possible solutions are:
$\left( {\frac{a_1}{a_2}} \middle/ {\frac{b_1}{b_2}} \right) $
Or, in case the brackets are not necessary:
$\left. {\frac{a_1}{a_2}} \middle/ {\frac{b_1}{b_2}} \right. $