A thin bi-convex lens made up of glass of refractive index `3//2` is placed is front of a plane mirror. The space between the lens and the mirror is filled with water of refractive index `4//3` . The radii of curvatures of the lens are `R_(1)=15cm` and `R_(2)=25 cm ` . A point object is placed at distance x from the surface whose curvature is `R_(1)`. The distance x is greater than the separation between the lens and the mirror. Find the value of x so that image must coincide with the object.

Explanation

To solve the problem, we need to find the distance \( x \) such that the image formed by the biconvex lens coincides with the object placed in front of it. Here’s a step-by-step solution:

### Step 1: Understand the System
We have a thin biconvex lens made of glass with a refractive index \( n_1 = \frac{3}{2} \) placed in front of a plane mirror. The space between the lens and the mirror is filled with water, which has a refractive index \( n_2 = \frac{4}{3} \). The radii of curvature of the lens are \( R_1 = 15 \, \text{cm} \) (for the surface facing the object) and \( R_2 = 25 \, \text{cm} \) (for the surface facing the mirror).

### Step 2: Apply the Lens Maker's Formula
The lens maker's formula for a thin lens is given by:

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