Explanation

(b) If all charges are in equilibrium, system is also in equilibrium.
Charge at centre : charge $q$ is in equilibrium because no net force acting on it corner charge :
If we consider the charge at corner $B.$ This charge will experience following forces
$F_{A} = k\frac{Q^{2}}{a^{2}},F_{C} = \frac{kQ^{2}}{a^{2}},F_{D} = \frac{kQ^{2}}{\left( a\sqrt{2} \right)^{2}}$ and $F_{O} = \frac{KQq}{\left( a\sqrt{2} \right)^{2}}$
Force at $B$ away from the centre $= F_{AC} + F_{D}$
$= \sqrt{F_{A}^{2} + F_{B}^{2}} + F_{D} = \sqrt{2}\frac{kQ^{2}}{a^{2}} + \frac{kQ^{2}}{{2a}^{2}} = \frac{kQ^{2}}{a^{2}}\left( \sqrt{2}\ + \frac{1}{2} \right)$
Force at $B$ towards the centre $= F_{O} = \frac{2kQq}{a^{2}}$
For equilibrium of charge at $B,\ F_{AC} + F_{D} = F_{O}$
$\Rightarrow \frac{kQ^{2}}{a^{2}}\left( \sqrt{2}\ + \frac{1}{2} \right) = \frac{2KQq}{a^{2}} \Rightarrow q = \frac{Q}{4}\left( 1 + 2\sqrt{2} \right)$