Convex side of a plano-convex lens is fully immersed in a liquid, of same refractive index (equal to ⁸⁄₅) as that of the lens. Which one among the following is correct regarding the image of a point source of light placed in air at 40 cm away from the lens ? (Assume the focal length of the plano-convex lens as 20 cm)

To solve this, we first determine the radius of curvature (R) of the plano-convex lens using the lens maker's formula in air: 1/f = (μ_l - 1)(1/R). Given f = 20 cm and μ_l = 8/5 (or 1.6), we get 1/20 = (1.6 - 1)/R, which gives R = 12 cm.

When the convex side is immersed in a liquid with the same refractive index (μ_liq = 1.6), the boundary between the lens and the liquid effectively disappears because there is no change in the refractive index. Thus, no refraction occurs at the curved surface. The system now behaves as a single plane interface between air (n₁ = 1) and the lens/liquid medium (n₂ = 1.6).

Using the refraction formula for a plane surface (R = ∞):
n₂/v - n₁/u = 0
1.6/v - 1/(-40) = 0
1.6/v = -1/40
v = -1.6 × 40 = -64 cm.

The image is formed 64 cm from the lens. In the context of this objective question, the magnitude 64 cm identifies option D as the correct choice.