A diverging lens of focal length `f_(1)` is placed in front of and coaxially with a concave mirror of foacl length `f_(2)`. Their separation is d. A parallel beam of light incident on the lens returns as a parallel beam from the arrangement. Then,

Options

A. the beam diameters of the incident and reflected beams must be the same
B. `d=2|f_(2)|-|f_(1)|`
C. `d=|f_(2)|-|f_(1)|`
D. if the entire arrrangement is immersed in water, the conditions will remain unaltered

Explanation

To solve the problem, we need to analyze the optical arrangement consisting of a diverging lens and a concave mirror. The goal is to determine the relationship between the focal lengths of the lens and mirror, as well as the separation between them, given that a parallel beam of light incident on the lens returns as a parallel beam after reflection.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a diverging lens (focal length \( f_1 \)) and a concave mirror (focal length \( f_2 \)).
- The distance between the lens and the mirror is \( d \).

2. **Ray Diagram**:
- Draw a ray diagram to visualize the situation. A parallel beam of light strikes the diverging lens.
- The lens will cause the rays to diverge. The diverging rays appear to come from a virtual focus on the same side as the incoming light.

3. **Finding the Virtual Focus of the Lens**:
- For a diverging lens, the focal length \( f_1 \) is negative.
- The virtual focus \( F_1 \) of the lens is located at a distance \( |f_1| \) from the lens on the same side as the incoming light.

4. **Reflected Rays from the Mirror**:
- The diverging rays from the lens will reach the concave mirror.
- The mirror will reflect these rays. The concave mirror has a focal length \( f_2 \) (positive value).

5. **Condition for Parallel Rays**:
- For the light to return as a parallel beam after reflection, the rays must converge at the focal point of the mirror.
- The distance from the mirror to the virtual focus \( F_1 \) must equal the focal length of the mirror, which means:
\[ d - |f_1| = f_2 \]

6. **Rearranging the Equation**:
- Rearranging the equation gives:
\[ d = f_2 + |f_1| \]

7. **Conclusion**:
- The separation \( d \) between the lens and the mirror is equal to the sum of the absolute value of the focal length of the diverging lens and the focal length of the concave mirror.

### Final Relationship:
\[ d = |f_1| + f_2 \]