A real image formed by a concave mirror is 4.5 times the size of the object. If the mirror is 20 cm from the object, its focallength is

Options

A. `90/11` cm
B. `120/11` cm
C. `150/11` cm
D. `180/11` cm

Explanation

To find the focal length of a concave mirror given that a real image is formed which is 4.5 times the size of the object, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Magnification**:
The magnification (M) for a mirror is given by the formula:
\[ M = -\frac{V}{U} \]
where \( V \) is the image distance and \( U \) is the object distance. Since the image is real and 4.5 times the size of the object, we have:
\[ M = -4.5 \]

2. **Identify the Object Distance**:
The object distance \( U \) is given as 20 cm. For mirrors, we take the object distance as negative:
\[ U = -20 \, \text{cm} \]

3. **Set Up the Magnification Equation**:
Substitute the values of \( M \) and \( U \) into the magnification formula:
\[ -4.5 = -\frac{V}{-20} \]
This simplifies to:
\[ 4.5 = \frac{V}{20} \]

4. **Calculate the Image Distance**:
Rearranging the equation to find \( V \):
\[ V = 4.5 \times 20 = 90 \, \text{cm} \]
Since the image is real, we take \( V \) as negative:
\[ V = -90 \, \text{cm} \]

5. **Apply the Mirror Formula**:
The mirror formula is given by:
\[ \frac{1}{F} = \frac{1}{V} + \frac{1}{U} \]
Substituting the values of \( V \) and \( U \):
\[ \frac{1}{F} = \frac{1}{-90} + \frac{1}{-20} \]

6. **Calculate the Right Side**:
Finding a common denominator (which is 180):
\[ \frac{1}{F} = -\frac{2}{180} - \frac{9}{180} = -\frac{11}{180} \]

7. **Find the Focal Length**:
Taking the reciprocal gives:
\[ F = -\frac{180}{11} \, \text{cm} \]

### Conclusion:
The focal length of the concave mirror is:
\[ F = -\frac{180}{11} \, \text{cm} \approx -16.36 \, \text{cm} \]