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, we need to analyze the behavior of a particle approaching a concave mirror and the relationship between the distance of the particle and its image.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a concave mirror with a focus (F) and a center of curvature (C).
- A particle (object) approaches the mirror from a very large distance (u = ∞).

2. **Initial Condition**:
- When the object is at infinity (u = ∞), the image is formed at the focus (V = F).
- The distance between the object and the image at this point is:
\[ \text{Distance} = |u - v| = |∞ - F| = ∞ \]

3. **Object at Center of Curvature**:
- As the object moves closer, let’s consider when it is at the center of curvature (u = R).
- The image is also formed at the center of curvature (v = R).
- The distance between the object and the image is:
\[ \text{Distance} = |u - v| = |R - R| = 0 \]
- Here, the distance decreases from infinity to zero.

4. **Object at Focus**:
- Next, consider when the object is at the focus (u = F).
- The image is formed at infinity (v = ∞).
- The distance between the object and the image is:
\[ \text{Distance} = |u - v| = |F - ∞| = ∞ \]
- The distance increases again.

5. **Object Inside the Focus**:
- If the object is moved inside the focus (u < F), the image formed is virtual and located behind the mirror.
- As the object approaches the mirror, the image moves closer to the object.
- For example, if the object is at the mirror's surface (u = 0), the image will also be at the same point (v = 0).
- The distance between the object and the image is:
\[ \text{Distance} = |u - v| = |0 - 0| = 0 \]
- Thus, the distance decreases again as the object moves towards the mirror.

6. **Conclusion**:
- The distance between the object and its image first decreases from infinity to zero as the object approaches the center of curvature, then increases to infinity when the object is at the focus, and finally decreases again to zero when the object is inside the focus.
- Therefore, the overall behavior of the distance between the particle and its image is: **first decreases, then increases, and then decreases again**.

### Final Answer:
The distance between the particle and its image will first decrease, then increase, and finally decrease again.