A plano-convex lens acts like a concave mirror of focal length 28 cm when its plane surface is silvered and like a concave mirror of focal length 10 cm when its curved surface is silvered. The refractive index of the material of the lens is

1.55

The correct answer is Option (2) → 1.55

Case-1: The plane surface of plano-convex lens is silvered

$\Rightarrow f_m=\infty, f_e=\frac{R}{\mu-1}, R \rightarrow$ radius of curvature of curved surface

$\frac{1}{f_{e q}}=\frac{-2}{f_e}+\frac{1}{\infty}$

$\Rightarrow f_{e q}=\frac{-R}{2(\mu-1)}$ and $\frac{-R}{2(\mu-1)}=-28$             ......(i)

Case-2: The spherical surface of plano-convex lens is silvered

$f_e=\frac{R}{\mu-1}, f_m=\frac{-R}{2}, \frac{1}{f_{e q}}=\frac{-2(\mu-1)}{R}-\frac{-2}{R}=\frac{-2 \mu}{R}$

$\Rightarrow f_{\text {eq }}=\frac{-R}{2 \mu} \text { and } \frac{-R}{2 \mu}=-10$             ......(ii)

Dividing (i) and (ii)

$=\frac{R}{\frac{2(\mu-1)}{\frac{R}{2 \mu}}}=\frac{28}{10} \Rightarrow \frac{\mu}{\mu-1}=\frac{14}{5} \Rightarrow 5 \mu=14 \mu-14$

$14=9 \mu \Rightarrow \mu=\frac{14}{9}$ = 1.55