A parallel beam of rays is incident on a convergent lens with a focal length of `40 cm.` Where should a divergent lens with a focal length of `15 cm` be placed for the beam of rays to remain parallel after passing through the two lenses.

Explanation

To solve the problem of where to place a divergent lens so that a parallel beam of rays remains parallel after passing through a convergent lens, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Focal length of the convergent lens (convex lens), \( f_1 = 40 \, \text{cm} \).
- Focal length of the divergent lens (concave lens), \( f_2 = -15 \, \text{cm} \) (negative because it is a divergent lens).

2. **Understand the Behavior of Light Through Lenses:**
- A parallel beam of rays incident on a convergent lens will converge at the focal point of the lens.
- The rays will meet at the focus of the convergent lens, which is located at a distance of \( 40 \, \text{cm} \) from the lens.

3. **Determine the Position of the Divergent Lens:**
- To keep the rays parallel after passing through both lenses, the divergent lens must be positioned such that it takes the rays that converge at the focal point of the convergent lens and makes them parallel again.
- The rays converge at the focal point of the convergent lens, which is at \( 40 \, \text{cm} \) from the convergent lens.

4. **Calculate the Distance Between the Lenses:**
- Let \( x \) be the distance from the convergent lens to the divergent lens.
- The rays will need to be \( 15 \, \text{cm} \) from the focus of the convergent lens to the divergent lens. This is because the focus of the divergent lens is \( 15 \, \text{cm} \) in front of it (since it is a divergent lens).
- Therefore, we can set up the equation:
\[ x + 15 = 40 \]

5. **Solve for \( x \):**
- Rearranging the equation gives:
\[ x = 40 - 15 = 25 \, \text{cm} \]

6. **Conclusion:**
- The divergent lens should be placed \( 25 \, \text{cm} \) behind the convergent lens for the rays to remain parallel after passing through both lenses.

### Final Answer:
The divergent lens should be placed **25 cm behind the convergent lens**.

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