A charge q is placed at the centre of the line joining two equal charges Q. The system of the three charges will be in equilibrium if q is equal to

Refer to Fig. 

The three charges will be in equilibrium, if no net force acts on each charge. The charge q is in equilibrium because the forces exerted on q by charge Q at A and charge Q at B are equal and opposite. The charge Q at A will be in equilibrium if the forces exerted on it by charge q and charge Q at B are equal and opposite, i.e. if 

$\frac{qQ}{4\pi {\epsilon }_0{r}^2}=-\frac{Q\times Q}{4\pi {\epsilon }_0(2r{)}^2}$

or $q=-\frac{Q}{4}$

Similarly, charge Q at B will be in equilibrium if q 

$=-\frac{Q}{4}$.