A convergent beam of light is incident on a convex mirror so as to converge to a distance 12 cm from the pole of the mirror. An inverted image of the same size is formed coincident with the virtual object. What is the focal length of the mirror ?

Options

A. 24 cm
B. 12 cm
C. 6 cm
D. 3 cm

Answer

To find the focal length of the convex mirror given the conditions of the problem, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- A convergent beam of light is incident on a convex mirror and converges to a distance of 12 cm from the pole of the mirror.
- An inverted image of the same size is formed coincident with the virtual object.

2. **Identifying the Image and Object Relationship**:
- Since the image is inverted and of the same size as the object, it indicates that the object is located at the center of curvature (C) of the mirror.

3. **Using the Mirror Formula**:
- For a convex mirror, the focal length (f) is related to the radius of curvature (R) by the formula:
\[ R = 2f \]

4. **Finding the Radius of Curvature**:
- Given that the distance from the pole of the mirror to the point where the light converges (which is effectively the center of curvature for the virtual object) is 12 cm, we can set:
\[ R = 12 \text{ cm} \]

5. **Calculating the Focal Length**:
- Now, substituting the value of R into the formula for focal length:
\[ f = \frac{R}{2} = \frac{12 \text{ cm}}{2} = 6 \text{ cm} \]

6. **Conclusion**:
- Therefore, the focal length of the convex mirror is:
\[ f = 6 \text{ cm} \]

### Final Answer:
The focal length of the convex mirror is **6 cm**.

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