Explanation

(d) Given : $(b - a) = 1 \times 10^{- 3}m$ …$(i)$
and $C = 4\pi\varepsilon_{0}\left( \frac{ab}{b - a} \right) = 1 \times 10^{- 6}$
$\Rightarrow 1 \times 10^{- 6} = \frac{1}{9 \times 10^{9}}\left( \frac{ab}{10^{- 3}} \right)$
$\Rightarrow ab = 9$ …$(ii)$
From equations (i) and (ii)
$b - \frac{9}{b} = \frac{1}{1000} \Rightarrow 1000\ b^{2} - b - 9000 = 0$
$\Rightarrow b = \frac{1 \pm \sqrt{( - 1)^{2} - 4(1000)( - 9000)}}{2 \times 1000}$
[Solving of quadratic equation]
$\Rightarrow b = \frac{1 \pm \sqrt{36 \times 10^{6}}}{2000} = \frac{\sqrt{36 \times 10^{6}}}{2000} = 3m$