A luminous object is placed 20 cm from surface of a convex mirror and a plane mirror is set so that virtual images formed in two mirrors coincide. If plane mirror is at a distance of 12cm from object. Then focal length of convex

Options

A. 20cm
B. 15cm
C. 10cm
D. 5cm

Explanation

To solve the problem, we need to find the focal length of the convex mirror given the distances of the object and the plane mirror. Let's break it down step by step.

### Step 1: Understand the Configuration
We have:
- A luminous object placed 20 cm from the surface of a convex mirror.
- A plane mirror is positioned such that the virtual images formed by both mirrors coincide.
- The plane mirror is 12 cm from the object.

### Step 2: Determine the Position of the Virtual Image from the Plane Mirror
For a plane mirror, the image distance (v) is equal to the object distance (u). Since the plane mirror is 12 cm away from the object, we have:
- \( u_{plane} = -12 \) cm (the negative sign indicates that the object is in front of the mirror).
- Therefore, the image distance for the plane mirror is:
\[ v_{plane} = -u_{plane} = 12 \text{ cm} \]

### Step 3: Determine the Position of the Virtual Image from the Convex Mirror
The total distance from the object to the convex mirror is 20 cm. Thus, the distance from the convex mirror to the virtual image formed by it must be calculated. The virtual image formed by the plane mirror is at 12 cm from the object, which means it is at:
\[ 20 \text{ cm} - 12 \text{ cm} = 8 \text{ cm} \]
from the convex mirror. Therefore, the image distance for the convex mirror is:
- \( v_{convex} = 8 \) cm (positive because the image is virtual and formed on the same side as the object).

### Step 4: Use the Mirror Formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
For the convex mirror:
- \( v = 8 \) cm (positive for virtual image),
- \( u = -20 \) cm (negative as per sign convention).

Substituting the values into the mirror formula:
\[ \frac{1}{f} = \frac{1}{8} + \frac{1}{-20} \]

### Step 5: Calculate the Focal Length
Now we need to find a common denominator to solve this:
\[ \frac{1}{f} = \frac{5}{40} - \frac{2}{40} = \frac{3}{40} \]
Thus,
\[ f = \frac{40}{3} \approx 13.33 \text{ cm} \]

### Conclusion
The focal length of the convex mirror is approximately \( 13.33 \) cm.

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