Explanation

(b) $\omega_{body} = 27\omega_{earth}$
$T^{2} \propto r^{3} \Rightarrow \omega^{2} \propto \frac{1}{r^{3}} \Rightarrow \omega \propto \frac{1}{r^{3/2}}\ \therefore\ r \propto \frac{1}{\omega^{2/3}}$
$\Rightarrow \frac{r_{body}}{r_{earth}} = \left( \frac{\omega_{earth}}{\omega_{body}} \right)^{2/3} = \left( \frac{1}{27} \right)^{2/3} = \frac{1}{9}$