Where would an object be placed in a medium of refractive index $\mathrm{\mu }$₁ , so that its real image is formed  equidistant from sphere of radius R and refractive index $\mathrm{\mu }$₂ which is also placed in the medium of refractive index $\mathrm{\mu }$₁ as shown in figure?

In order to form the image at the same distance, the ray should emerge out of the sphere at the same angle as its incident angle. To obtain this, the ray should be parallel to the axis of the sphere as shown in the figure.

$\frac{{\mu }_2}{+\mathrm{\infty }}-\frac{{\mu }_1}{-\mathrm{x}}=\frac{{\mu }_2-{\mu }_1}{\mathrm{R}}$

$x=\frac{{\mu }_{1}R}{{\mu }_2-{\mu }_1}$