Hint: The apparent depth of the disc depends on the refractive index of the medium.

Step 1: Draw the diagram and find the value of the angle of incident and the angle of refraction.

Referring to the figure, AM is the direction of the incidence ray before the liquid is filled. After the liquid is filled in, BM is the direction of the incident ray. Refracted ray in both cases is the same as that along with AN.

Let the disc is placed at a distance d from O as shown in the figure.
                                          
Also, considering angle, $\angle \mathrm{ANM}$=90$°$, OM=a, BP=CP-BP=a-R, AP=a+R
Here, in the figure,
                                       $\mathrm{sini}=\frac{\mathrm{a}-\mathrm{R}}{\sqrt{{\mathrm{d}}^2+{(\mathrm{a}-\mathrm{R})}^2}}$
and,                                 $\sin \mathrm{\alpha }= \cos (90-\mathrm{\alpha })=\frac{\mathrm{a}+\mathrm{R}}{\sqrt{{\mathrm{d}}^2+{(\mathrm{a}+\mathrm{R})}^2}}$

Step 2: Find the value of the refractive index.
By applying Snell's law,
                                             $\frac{1}{\mathrm{\mu }}=\frac{\sin \mathrm{i}}{\sin \mathrm{r}}=\frac{\sin \mathrm{i}}{\sin \mathrm{\alpha }}$

Step 3: Find the position of the disc.
On substituting the values, we have the separation,
                                               $\mathrm{d}=\frac{\mathrm{\mu }({\mathrm{a}}^2-{\mathrm{b}}^2)}{\sqrt{{(\mathrm{a}+\mathrm{r})}^2-\mathrm{\mu }{(\mathrm{a}-\mathrm{r})}^2}}$
This is the required expression.