A point charge Q is located on the axis of a disc of radius $R$ at a distance $b$ from the plane of the disk. If one-fourth of the electric flux from the charge passes through the disk, then $R=\sqrt{k}b$.  Find $k$.

Total flux associated with charge Q for solid angle $4\pi$  is $\frac{Q}{{\epsilon }_0}.$  Solid angle subtended by the disc at Q is

$\Omega =2\pi \left(1-\cos \alpha \right)$

$=2\pi \left(1-\frac{b}{\sqrt{b^2+R^2}}\right)$

Electric flux through the disc $=\frac{Q\Omega }{{\epsilon }_{0}4\pi }.$

It is given to be $\frac{Q}{4{\epsilon }_0}$

So  $\frac{Q}{4{\epsilon }_0}=\frac{Q\Omega }{{\epsilon }_0\pi }.$ solving we get, $\Omega =\mathrm{\pi }, \Rightarrow \mathrm{cos\alpha }=\frac{1}{2}$

 $\frac{b}{\sqrt{b^2+R^2}}=\frac{1}{2}$ $\Rightarrow R=\sqrt{3}b$