Two point charges +4q and +q are placed at a distance L apart. A third charge Q is so placed that all the three charges are in equilibrium. What is the location and magnitude of third charge?

Lat third charge be placed at a distance ${\mathrm{x}}_1$ from +4q charge as shown and it should negative 

For neutral point between 4q and q, 

$\frac{\mathrm{k}4\mathrm{q}}{{\mathrm{x}}_1^2}=\frac{\mathrm{kq}}{{\left(\mathrm{L}-{\mathrm{x}}_1\right)}^2}$

$$\begin{gathered} \text{ Now, }{\mathrm{x}}_1=\frac{\mathrm{L}}{1+\sqrt{\frac{\mathrm{q}}{4\mathrm{q}}}}=\frac{2\mathrm{L}}{3},{\mathrm{x}}_2=\frac{\mathrm{L}}{3} \\ \text{ For equilibrium of }\mathrm{q},\frac{\mathrm{k}4\mathrm{q}}{{\mathrm{L}}^2}=\frac{\mathrm{kQ}}{{\mathrm{x}}_2^2} \\ \Rightarrow \mathrm{Q}=4\mathrm{q}{\left(\frac{\mathrm{L}/3}{\mathrm{L}}\right)}^2=\frac{4\mathrm{q}}{9}\Rightarrow \mathrm{Q}=-\frac{4\mathrm{q}}{9} \end{gathered}$$