A particle approaches from very large distance towards concave mirror along the principal axis. By the time the particle reaches the mirror the distance between the particle and its image will

Options

A. first increases then decreases
B. first decreases then increases
C. first decreases then decreases then again increases
D. first decreases then increases then again decrease

Explanation

To solve the problem of determining the distance between a particle and its image as the particle approaches a concave mirror, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Initial Condition**:
- The particle is initially at a very large distance from the concave mirror, which can be considered as being at infinity.

2. **Image Formation at Infinity**:
- When the object (particle) is at infinity, the image formed by the concave mirror is at the focus (F) of the mirror. The distance between the object and the image at this point is effectively infinite.

3. **Approaching the Mirror**:
- As the particle moves closer to the mirror, the image will also move. The image will start moving from the focus towards the center of curvature (C) as the object approaches the mirror.

4. **At the Center of Curvature**:
- When the particle reaches the center of curvature (C), the image will also be formed at the center of curvature (C). At this point, the distance between the object and the image is zero because they occupy the same position.

5. **Moving Towards the Mirror (Pole)**:
- As the particle continues to move towards the mirror (beyond C towards the pole), the image will start to move further away from the mirror (towards infinity). Thus, the distance between the particle and its image will start to increase.

6. **Final Position**:
- If the particle reaches the pole (the surface of the mirror), the image will be formed at a point that is very far away, effectively at infinity. Therefore, the distance between the particle and its image will again be very large.

### Summary of Distance Changes:
- Initially, the distance is infinite (particle at infinity).
- As the particle approaches, the distance decreases until it reaches C (distance = 0).
- After passing C, the distance starts to increase again as the particle approaches the pole.

### Conclusion:
The distance between the particle and its image first decreases to zero when the particle is at C and then increases again as the particle moves closer to the pole.