By Kirchhoff's rule, energy radiated by planet is proportional to fourth power of its temperature.

$\Rightarrow \quad E_{1}=k T^{4}=k\left(f T^{\prime}\right)^{4}$

where, temperature of planet is $f T^{\prime}$ (given). Energy received by planet

= Solar constant of planet

$=E_{2}=\frac{R^{2} \sigma\left(T^{\prime}\right)^{4}}{d^{2}}$

where, $R=$ radius of star,

$T^{\prime}=$ temperature of star,

$\sigma=$ Stefan's constant

and $d=$ distance of planet and star.

In equilibrium,

$E_{1}=E_{2} \Rightarrow k f^{4}\left(T^{\prime}\right)^{4}=\sigma\left(\frac{R^{2}}{d^{2}}\right) \cdot T^{\prime 4}$

$\Rightarrow \quad f \propto \sqrt{\frac{R}{d}}$