The dispersive powers of glasses of lenses used in an achromatic pair are in the ratio 5 : 3. If the focal length of the concave lens is 15 cm , then the nature and focal length of the other lens would be

Options

A. convex, 9 cm
B. concave, 9 cm
C. convex, 25 cm
D. concave, 25 cm

Explanation

To solve the problem, we need to determine the nature and focal length of the other lens in an achromatic pair, given the dispersive powers and the focal length of the concave lens.

### Step-by-Step Solution:

1. **Understand the Relationship**:
The dispersive powers (ω) of the lenses in an achromatic pair are related to their focal lengths (f). The relationship is given by:
\[ \frac{\omega_1}{\omega_2} = \frac{f_1}{f_2} \]
where:
- \( \omega_1 \) is the dispersive power of the concave lens,
- \( \omega_2 \) is the dispersive power of the other lens,
- \( f_1 \) is the focal length of the concave lens,
- \( f_2 \) is the focal length of the other lens.

2. **Substitute Given Values**:
From the problem, we know:
- The ratio of dispersive powers is \( \frac{5}{3} \),
- The focal length of the concave lens \( f_1 = -15 \) cm (negative because it is a concave lens).

Substituting these values into the equation gives:
\[ \frac{5}{3} = \frac{-15}{f_2} \]

3. **Cross-Multiply to Solve for \( f_2 \)**:
Cross-multiplying gives:
\[ 5f_2 = -45 \]
Now, solving for \( f_2 \):
\[ f_2 = \frac{-45}{5} = -9 \text{ cm} \]

4. **Determine the Nature of the Lens**:
Since \( f_2 \) is negative, this indicates that the other lens is also a concave lens.

### Final Answer:
- The nature of the other lens is **concave**.
- The focal length of the other lens is **-9 cm**.