An object is placed at a distance of `36 cm` from a convex mirror. A plane mirror is placed inbetween so that the two virtual images so formed coincide. If the plane mirror is at a distance of `24 cm` from the object, find the radius of curvature of convex mirror.

Explanation

To solve the problem step by step, we will use the mirror formula and the information provided in the question.

### Step 1: Identify the distances
- The object distance from the convex mirror (u) is given as \(36 \, \text{cm}\). Since the object is in front of the mirror, we take this value as negative:
\[ u = -36 \, \text{cm} \]
- The distance of the plane mirror from the object is \(24 \, \text{cm}\). Therefore, the distance from the convex mirror to the plane mirror is:
\[ d = 36 \, \text{cm} - 24 \, \text{cm} = 12 \, \text{cm} \]

### Step 2: Determine the image distance from the convex mirror
- The virtual image formed by the plane mirror is at the same distance as the object from the plane mirror, which is \(24 \, \text{cm}\) behind the plane mirror. Since the distance from the convex mirror to the plane mirror is \(12 \, \text{cm}\), the image formed by the convex mirror coincides with the image formed by the plane mirror. Thus, the image distance (v) from the convex mirror is:
\[ v = +12 \, \text{cm} \]

### Step 3: Use the mirror formula to find the focal length
- The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
- Substituting the values of \(v\) and \(u\):
\[ \frac{1}{f} = \frac{1}{12} + \frac{1}{-36} \]
- Finding a common denominator (which is \(36\)):
\[ \frac{1}{f} = \frac{3}{36} - \frac{1}{36} = \frac{2}{36} \]
- Simplifying gives:
\[ \frac{1}{f} = \frac{1}{18} \]
- Thus, the focal length \(f\) is:
\[ f = 18 \, \text{cm} \]

### Step 4: Calculate the radius of curvature
- The radius of curvature \(R\) is related to the focal length by the formula:
\[ R = 2f \]
- Substituting the value of \(f\):
\[ R = 2 \times 18 \, \text{cm} = 36 \, \text{cm} \]

### Final Answer
The radius of curvature of the convex mirror is:
\[ \boxed{36 \, \text{cm}} \]