Find the nature and focal length of a lens which must be placed in contact with a concave lens of focal length`25cm` in order that the lens combination may produce a real image of an object placed at infinity.

Answer

To solve the problem of finding the nature and focal length of a lens that must be placed in contact with a concave lens of focal length -25 cm (since the focal length of a concave lens is negative) to produce a real image of an object placed at infinity, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Focal Length of the Concave Lens:**
The focal length of the concave lens is given as \( f_1 = -25 \, \text{cm} \).

2. **Understand the Requirement for a Real Image:**
To produce a real image from an object at infinity, the combination of the lenses must have a net positive focal length. This means that the converging effect of the convex lens must outweigh the diverging effect of the concave lens.

3. **Let the Focal Length of the Convex Lens be \( f_2 \):**
We will denote the focal length of the convex lens as \( f_2 \). Since it is a convex lens, \( f_2 \) will be positive.

4. **Use the Lens Formula for Combination:**
The formula for the effective focal length \( f \) of two lenses in contact is given by:
\[ \frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} \]
Substituting \( f_1 = -25 \, \text{cm} \):
\[ \frac{1}{f} = \frac{1}{-25} + \frac{1}{f_2} \]

5. **Condition for a Real Image:**
For the combination to produce a real image, \( f \) must be positive. Therefore, we need:
\[ \frac{1}{f_2} - \frac{1}{25} > 0 \]
This simplifies to:
\[ \frac{1}{f_2} > \frac{1}{25} \]
or
\[ f_2 < 25 \, \text{cm} \]

6. **Conclusion on the Nature of the Lens:**
Since \( f_2 \) must be positive and less than 25 cm, the nature of the lens is a **convex lens** with a focal length of less than 25 cm.

### Summary of Findings:
- The nature of the lens is **convex**.
- The focal length \( f_2 \) must be **less than 25 cm**.