Physics - Geometrical Optics Question with Solution | TestHub

For the final image to coincide with the object, the rays must retrace their path, so after refraction at the first surface they must strike the silvered surface normally - the first surface must form the image at the centre of curvature of the silvered surface, 20 cm to the left of the lens.

First surface (air to glass, R = +10 cm), v = -20 cm, u = -x:

$\frac{{\mu }_2}{v}-\frac{{\mu }_1}{u}=\frac{{\mu }_2-{\mu }_1}{R}\Rightarrow \frac{3/2}{-20}-\frac{1}{-x}=\frac{1/2}{10}$

$-\frac{3}{40}+\frac{1}{x}=\frac{1}{20}\Rightarrow \frac{1}{x}=\frac{1}{8}\Rightarrow x=8\text{cm}$

Answer: 8