A thin plano convex lens made of glass of refractive index 1.5 is immersed in a liquid of refractive index 1.2. When the plane side of the lens is silver coated for complete reflection, the lens immersed in the liquid behaves like a concave mirror of focal length 0.2 m. The radius of curvature of the curved surface of the lens is

For the plano-convex lens immersed in liquid:

$$\mu_g = 1.5, \quad \mu_l = 1.2, \quad R_1 = R, \quad R_2 = \infty$$

$$P_l = \frac{1}{f_l} = \left(\frac{\mu_g}{\mu_l} - 1\right)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \implies P_l = \left(\frac{1.5}{1.2} - 1\right)\left(\frac{1}{R} - \frac{1}{\infty}\right) = \frac{0.25}{R} = \frac{1}{4R}$$

For the plane silvered surface acting as a plane mirror:

$$R_m = \infty \implies P_m = -\frac{1}{f_m} = -\frac{1}{\infty} = 0$$

Equivalent power of silvered system behaving as a concave mirror:

$$F_{\text{eq}} = -0.2\text{ m}$$

$$P_{\text{eq}} = -\frac{1}{F_{\text{eq}}} = 2P_l + P_m \implies -\frac{1}{-0.2} = 2\left(\frac{1}{4R}\right) + 0 \implies 5 = \frac{1}{2R} \implies R = \frac{1}{10} = 0.10\text{ m}$$