Electric Charge & Coulomb's Law question from MHT CET 2024 (3 May, Shift 2)

Answer: (d)

1. Let the side be $a$. Repulsion of $+Q$ by the other $+Q$ (distance $\sqrt{2}a$): $\dfrac{kQ^2}{2a^2}$ along the diagonal, outward.
2. Attraction by each $-q$ (distance $a$): $\dfrac{kQq}{a^2}$; the two are perpendicular, resultant $\dfrac{\sqrt{2}kQq}{a^2}$ along the diagonal, inward.
3. Balance: $\dfrac{kQ^2}{2a^2} = \dfrac{\sqrt{2}kQq}{a^2} \Rightarrow Q = 2\sqrt{2}\,q$.
4. So $\dfrac{+Q}{-q} = -2\sqrt{2}$, answer (d).