Answer
$\frac{q}{Q} = 0.707 $
Work Step by Step
From the question,
$q_1 = +Q$
$q_2 = q_3 = q $
$q_4 = -2.00Q$
To find the $q/Q$ when the net force in particle a is 0, we need to equate these equations.
$\frac{kQq}{a^2} = \frac{k Q |2Q| }{(\sqrt{2}a)^2} cos 45^o $
$\frac{kQq}{a^2} = \frac{2kQ^2}{(\sqrt2 a)^2} \times \frac{\sqrt{2}}{2}$
$q = \frac{2kQ^2a^2}{(\sqrt2 a)^2 Q k} \times \frac{\sqrt{2}}{2}$
$\frac{q}{Q} = \frac{2kQa^2}{2 a^2 Q k} \times \frac{\sqrt{2}}{2} $
Now simplify all these
$\frac{q}{Q} = \frac{\sqrt{2}}{2} $
$\frac{q}{Q} = 0.707 $
