An object placed at 20cm in front of a concave mirror produces three times magnified real image. What is the focal length of the concave mirror?

Options

A. 15cm
B. 6.6cm
C. 10cm
D. 7.5cm

Explanation

To solve the problem step by step, we will use the mirror formula and the magnification formula for concave mirrors.

### Step 1: Understand the given information
- The object distance (u) is given as 20 cm. Since the object is in front of the mirror, we take it as negative:
\[ u = -20 \, \text{cm} \]
- The magnification (m) is given as 3 times, and since it is a real image, it will be negative:
\[ m = -3 \]

### 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.

Substituting the known values into the magnification formula:
\[ -3 = -\frac{v}{-20} \]

### Step 3: Solve for the image distance (v)
From the equation:
\[ -3 = \frac{v}{20} \]
Multiplying both sides by 20:
\[ v = -3 \times 20 \]
\[ v = -60 \, \text{cm} \]

### Step 4: Use the mirror formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Where:
- \( f \) is the focal length of the mirror.

Substituting the values of \( v \) and \( u \):
\[ \frac{1}{f} = \frac{1}{-60} + \frac{1}{-20} \]

### Step 5: Calculate the right-hand side
Finding a common denominator (which is 60):
\[ \frac{1}{f} = \frac{-1}{60} + \frac{-3}{60} \]
\[ \frac{1}{f} = \frac{-4}{60} \]
\[ \frac{1}{f} = \frac{-1}{15} \]

### Step 6: Find the focal length (f)
Taking the reciprocal:
\[ f = -15 \, \text{cm} \]

### Conclusion
The focal length of the concave mirror is:
\[ f = -15 \, \text{cm} \]