A ball at O is in equilibrium as it is attached with two strings AO and DO, which are tied at A and D. AO = DO = $\mathrm{a}\sqrt{5}$. The charges at A, B, C, and, D are +q, +Q, +2Q, and -q, respectively. Find the correct options. The ball at O is positively charged.

For ball O (positively charged) to be in equilibrium at the middle of BC

$\left[\frac{\mathrm{q}}{4{\mathrm{\pi \epsilon }}_0(\mathrm{a}\sqrt{5}{)}^2}+\frac{\mathrm{q}}{4{\mathrm{\pi \epsilon }}_0(\mathrm{a}\sqrt{5}{)}^2}\right]\cos \mathrm{\theta }+\frac{\mathrm{Q}}{4{\mathrm{\pi \epsilon }}_0{\mathrm{a}}^2}$

$=\frac{2\mathrm{Q}}{4{\mathrm{\pi \epsilon }}_0{\mathrm{a}}^2}$

$\mathrm{q}=\frac{5\sqrt{5}\mathrm{Q}}{2}\left(\text{ as }\cos \mathrm{\theta }=\frac{\mathrm{a}}{\mathrm{a}\sqrt{5}}\right)$

If $\mathrm{q}=5\sqrt{5}\mathrm{Q}/2$, ball will be in equilibrium, even if charge are interchanged. (1) is wrong and (3) and (4) are correct. If charge at C is +Q, then net horizontal force on the charge at O due to the charges at A and D will not be zero and hence the at O will not be in equilibrium.Therefore $\left(2\right) is wrong .$