Explanation

(a) When plates of capacitor are separated by a dielectric medium of dielectric constant $K$, its capacity
$C_{m} = \frac{K\varepsilon_{0}A}{d} = KC_{0}$
$ie,\ \ \ \ \ \ \ \ C_{m} = KC_{0}$
Here,$\ \ \ \ C_{0} = C$
$\therefore\ \ \ \ \ \ \ \ \ C_{m} = KC$
Now, two capacitors of capacities $KC$ and $C$ are in series, their effective capacitance
$\frac{1}{C^{'}} = \frac{1}{KC} + \frac{1}{C}$
or $\ \ \ \ \ \frac{1}{C^{'}}\ = \frac{1 + K}{KC}$
$\therefore\ C^{'} = \frac{KC}{K + 1}$