b
(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
\({F_A} = k\frac{{{Q^2}}}{{{a^2}}},\)\({F_C} = \frac{{k{Q^2}}}{{{a^2}}}\),\({F_D} = \frac{{k{Q^2}}}{{{{(a\sqrt 2 )}^2}}}{\rm{and}}\,{F_O} = \frac{{KQq}}{{{{(a\sqrt 2 )}^2}}}\)
Force at \(B\) away from the centre = \({F_{AC}} + {F_D}\)
\( = \sqrt {F_A^2 + F_C^2} + {F_D} = \sqrt 2 \frac{{k{Q^2}}}{{{a^2}}} + \frac{{k{Q^2}}}{{2{a^2}}} = \frac{{k{Q^2}}}{{{a^2}}}\left( {\sqrt 2 + \frac{1}{2}} \right)\)
Force at \(B\) towards the centre \( = {F_O} = \frac{{2kQq}}{{{a^2}}}\)
For equilibrium of charge at \(B\), \({F_{AC}} + {F_D} = {F_O}\)
\(==>\) \(\frac{{K{Q^2}}}{{{a^2}}}\left( {\sqrt 2 + \frac{1}{2}} \right) = \frac{{2KQq}}{{{a^2}}}\) \(==>\) \(q = \frac{Q}{4}\left( {1 + 2\sqrt 2 } \right)\)