A converging lens of focal length `f_(1)` is placed in front of and coaxially with a convex mirror of focal 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=f_(1)-2|f_(2)|`
C. `d=f_(1)-|f_(2)|`
D. if the entire arrangement is immersed in water, the conditions will remain unaltered

Explanation

To solve the problem, we need to analyze the arrangement of the converging lens and the convex mirror, and understand the conditions under which a parallel beam of light incident on the lens returns as a parallel beam after reflection from the mirror.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a converging lens with focal length \( f_1 \).
- A convex mirror with focal length \( f_2 \) is placed coaxially behind the lens.
- The distance between the lens and the mirror is \( d \).

2. **Condition for Parallel Beams**:
- A parallel beam of light incident on the lens will converge to its focal point \( f_1 \).
- After passing through the lens, the rays will converge at a point located at a distance \( f_1 \) from the lens.

3. **Reflection from the Convex Mirror**:
- The rays that converge at the focal point \( f_1 \) will then strike the convex mirror.
- For the rays to return as a parallel beam after reflection from the convex mirror, they must behave as if they are coming from the focal point of the mirror.

4. **Using the Focal Length of the Mirror**:
- The focal length \( f_2 \) of the convex mirror is defined as the distance from the mirror to the point where parallel rays converge after reflection.
- The distance from the lens to the mirror is \( d \), so the effective distance from the focal point of the lens to the mirror is \( d - f_1 \).

5. **Setting Up the Equation**:
- For the rays to retrace their path and return as parallel, the distance \( d - f_1 \) must equal \( f_2 \) (the focal length of the mirror).
- Therefore, we can write the equation:
\[ d - f_1 = f_2 \]
- Rearranging gives:
\[ d = f_1 + f_2 \]

6. **Considering the Object Distance for the Mirror**:
- The object distance for the mirror when the rays converge at the focal point is \( 2f_2 \) (since the object must be at twice the focal length for the rays to return parallel).
- Thus, we have another relationship:
\[ d = 2f_2 \]

7. **Equating the Two Expressions for \( d \)**:
- From the two equations we derived:
\[ f_1 + f_2 = 2f_2 \]
- Simplifying gives:
\[ f_1 = f_2 \]

8. **Conclusion**:
- The conditions for the parallel beam to return parallel after reflection are satisfied when \( d = f_1 + f_2 \) and \( f_1 = f_2 \).

### Final Answer:
The relationship between the focal lengths and the separation distance is:
\[ d = f_1 + f_2 \]