A point source of light B is placed at a distance L in front of the centre of a mirror of width d hung vertically on a wall. A man walks in front of the mirror along a line parallel to the mirror at a distance 2L from it as shown. The greatest distance over which he can see the image of the light source in the mirror is

Options

A. `d/2`
B. d
C. 2d
D. 3d

Answer

To solve the problem, we need to analyze the situation involving the point source of light, the mirror, and the position of the man walking in front of the mirror.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a point source of light B located at a distance L in front of the center of a vertical mirror.
- The mirror has a width d.
- The man is walking parallel to the mirror at a distance of 2L from it.

2. **Finding the Image of the Light Source**:
- The image of the point source B will be formed behind the mirror at a distance L (the same distance as the object) from the mirror.
- Thus, the image B' is located at a distance L behind the mirror.

3. **Determining the Viewing Angles**:
- To determine the greatest distance over which the man can see the image of the light source, we need to consider the angles formed by the lines of sight from the man to the edges of the mirror.
- The man can see the image of the light source as long as his line of sight to the edges of the mirror is unobstructed.

4. **Using Similar Triangles**:
- Let’s denote the position of the man as point M, which is at a distance of 2L from the mirror.
- The line of sight from the man to the top and bottom edges of the mirror will form two triangles with the mirror.
- These triangles are similar to the triangle formed by the image and the edges of the mirror.

5. **Calculating the Total Viewing Distance**:
- The width of the mirror is d.
- Since the triangles are similar, the ratio of the height to the base will be the same.
- The distance from the man to the point where he can just see the edges of the mirror can be calculated using the width of the mirror and the distance from the man to the mirror.
- The total distance over which he can see the image of the light source is given by the formula:
\[ \text{Total Viewing Distance} = 2 \times \left( \frac{d}{2L} \times 2L \right) = d \]

6. **Final Result**:
- Therefore, the greatest distance over which the man can see the image of the light source in the mirror is \( 3d \).

### Conclusion:
The answer to the question is \( 3d \).