Explanation

(a) Given electric potential of spheres are same $ie,\ $
$V_{A} = V_{B}$
$\frac{1}{4\pi\varepsilon_{0}}\frac{\theta_{1}}{a} = \frac{1}{4\pi\varepsilon_{0}}.\frac{\theta_{2}}{b}\ $
$\frac{\theta_{1}}{\theta^{2}} = \frac{a}{b}\ \ \ \ \ \ \ \ldots\ldots\ldots.(i)$
A surface charge density
$\sigma = \frac{a}{4\pi r^{2}}$
$\Longrightarrow \frac{\sigma_{1}}{\sigma_{2}} = \frac{\theta_{1}}{\theta_{2}} \times \frac{b^{2}}{b^{2}}$
$\ \ \ \ \ \ \ \ \ \ \ \ = \frac{a}{b} \times \frac{b^{2}}{a^{2}} = \frac{b}{a}$