A man uses a concave mirror for shaving. He keeps his face at a distance of `20cm` form the mirror and gets an image which is `1.5` times entarged. Find the focal length of the mirror.

Explanation

To find the focal length of the concave mirror used by the man for shaving, we can follow these steps:

### Step 1: Identify the given values
- Object distance (u) = -20 cm (the distance is negative because it is measured against the direction of the incoming light for mirrors)
- Magnification (m) = 1.5 (the image is enlarged)

### Step 2: Use the magnification formula
The magnification (m) for mirrors is given by the formula:
\[ m = -\frac{v}{u} \]
Where:
- \( v \) is the image distance
- \( u \) is the object distance

Substituting the known values into the magnification formula:
\[ 1.5 = -\frac{v}{-20} \]
This simplifies to:
\[ 1.5 = \frac{v}{20} \]

### Step 3: Solve for image distance (v)
To find \( v \), we can rearrange the equation:
\[ v = 1.5 \times 20 \]
\[ v = 30 \, \text{cm} \]
(Note: The image distance is positive because the image is formed on the same side as the object for a concave mirror when the object is within the focal length.)

### Step 4: Use the mirror formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the values of \( v \) and \( u \):
\[ \frac{1}{f} = \frac{1}{30} + \frac{1}{-20} \]

### Step 5: Calculate the focal length (f)
To find \( f \), we need to find a common denominator:
- The least common multiple (LCM) of 30 and 20 is 60.

Rewriting the fractions:
\[ \frac{1}{f} = \frac{2}{60} - \frac{3}{60} \]
\[ \frac{1}{f} = \frac{-1}{60} \]

Now, taking the reciprocal:
\[ f = -60 \, \text{cm} \]

### Conclusion
The focal length of the concave mirror is \( -60 \, \text{cm} \).

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