A man uses a concave mirror for shaving. He keeps his face at a distance of 25 cm from the mirror and gets an image which is 1.4 times enlarged. Find the focal length of the mirror

Explanation

To solve the problem of finding the focal length of a concave mirror used by a man for shaving, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values:**
- Object distance (U) = -25 cm (negative because the object is in front of the mirror)
- Magnification (M) = 1.4 (the image is enlarged)

2. **Relate Magnification to Image Distance:**
- The magnification formula is given by:
\[ M = -\frac{V}{U} \]
- Rearranging this gives:
\[ V = -M \cdot U \]
- Substituting the values:
\[ V = -1.4 \cdot (-25) = 1.4 \cdot 25 = 35 \, \text{cm} \]
- Thus, the image distance (V) = +35 cm (positive because the image is formed on the same side as the object for a concave mirror).

3. **Use the Mirror Formula:**
- The mirror formula is:
\[ \frac{1}{F} = \frac{1}{V} + \frac{1}{U} \]
- Substituting the values of V and U:
\[ \frac{1}{F} = \frac{1}{35} + \frac{1}{-25} \]

4. **Calculate the Right Side:**
- Finding a common denominator (which is 175):
\[ \frac{1}{F} = \frac{5}{175} - \frac{7}{175} = \frac{5 - 7}{175} = \frac{-2}{175} \]

5. **Find the Focal Length (F):**
- Taking the reciprocal gives:
\[ F = -\frac{175}{2} = -87.5 \, \text{cm} \]
- The negative sign indicates that it is a concave mirror.

### Final Answer:
The focal length of the concave mirror is **-87.5 cm**.

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