In an experiment to find the focal lengh of a concave mirror, an object is palced at a distance of 40 cm from it and forms image at 24 cm from the mirror, ten the focal length is

Options

A. `-15` cm
B. `-16` cm
C. `-30` cm
D. `-32` cm

Answer

To find the focal length of a concave mirror using the given distances, we can use the mirror formula:

\[ \frac{1}{F} = \frac{1}{V} + \frac{1}{U} \]

Where:
- \( F \) is the focal length of the mirror,
- \( V \) is the image distance,
- \( U \) is the object distance.

### Step-by-Step Solution:

1. **Identify the given values**:
- Object distance \( U = -40 \, \text{cm} \) (negative because it is measured against the direction of incident light).
- Image distance \( V = -24 \, \text{cm} \) (negative because the image is formed on the same side as the object for a concave mirror).

2. **Apply the mirror formula**:
\[ \frac{1}{F} = \frac{1}{V} + \frac{1}{U} \]

3. **Substitute the values into the formula**:
\[ \frac{1}{F} = \frac{1}{-24} + \frac{1}{-40} \]

4. **Find a common denominator**:
The common denominator for 24 and 40 is 120. Rewrite the fractions:
\[ \frac{1}{-24} = \frac{-5}{120}, \quad \frac{1}{-40} = \frac{-3}{120} \]

5. **Combine the fractions**:
\[ \frac{1}{F} = \frac{-5 - 3}{120} = \frac{-8}{120} \]

6. **Simplify the fraction**:
\[ \frac{1}{F} = \frac{-2}{30} = \frac{-1}{15} \]

7. **Invert to find \( F \)**:
\[ F = -15 \, \text{cm} \]

### Final Answer:
The focal length of the concave mirror is \( F = -15 \, \text{cm} \).

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