A concave mirrorr of focal length `f_(1)` is placed at a distance of `d` from a convex lens of focal length `f_(2)`. A beam of light coming from infinity and falling on this convex lens-concave mirrorr combination returns to infinity. The distance `d` must equal.

Options

A. `f_(1)+f_(2)`
B. `-f_(1)+f_(2)`
C. `2f_(1)+f_(2)`
D. `-2f_(1)+f_(2)`

Explanation

To solve the problem, we need to analyze the optical system consisting of a convex lens and a concave mirror. We want to find the distance \( d \) between the lens and the mirror such that a beam of light coming from infinity returns to infinity after reflecting off the mirror.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a convex lens with a focal length \( f_2 \).
- A concave mirror with a focal length \( f_1 \) is placed at a distance \( d \) from the lens.
- A beam of light coming from infinity is incident on the lens.

2. **Light Ray Behavior**:
- When parallel rays (from infinity) pass through the convex lens, they converge at the focal point of the lens. The distance from the lens to the focal point is \( f_2 \).

3. **Image Formation by the Lens**:
- The image formed by the lens will be at a distance \( f_2 \) from the lens on the opposite side. This image acts as a virtual object for the concave mirror.

4. **Position of the Mirror**:
- The concave mirror is placed at a distance \( d \) from the lens. Therefore, the distance from the image formed by the lens to the mirror is \( d - f_2 \).

5. **Condition for Returning to Infinity**:
- For the light rays to return to infinity after reflecting off the concave mirror, the virtual object (the image formed by the lens) must be at the center of curvature of the mirror. The center of curvature is located at a distance of \( 2f_1 \) from the mirror.

6. **Setting Up the Equation**:
- The distance from the mirror to the virtual object (image formed by the lens) must equal the distance to the center of curvature:
\[ d - f_2 = 2f_1 \]

7. **Solving for \( d \)**:
- Rearranging the equation gives:
\[ d = 2f_1 + f_2 \]

### Final Answer:
The distance \( d \) must equal \( 2f_1 + f_2 \).

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