A convex mirror forms an image one-fourth the size of the object. If object is at a distance of 0.5 m from the mirror the focal length of the mirror is

Options

A. 0.16
B. `-1.5 m`
C. 0.4 m
D. `-0.4 m`

Answer

To find the focal length of a convex mirror given that the image is one-fourth the size of the object and the object is placed at a distance of 0.5 m from the mirror, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Object distance (U) = -0.5 m (negative because it is a convex mirror)
- Image size is one-fourth the object size, which means the magnification (m) = HI/HO = 1/4.

2. **Relate Magnification to Image and Object Distances:**
- The magnification formula for mirrors is given by:
\[ m = -\frac{B}{U} \]
- Here, HI is the height of the image, HO is the height of the object, B is the image distance, and U is the object distance.

3. **Substituting the Magnification:**
- Since \( m = \frac{1}{4} \), we can write:
\[ \frac{1}{4} = -\frac{B}{-0.5} \]
- Rearranging gives:
\[ B = \frac{0.5}{4} = 0.125 \text{ m} \]

4. **Use the Mirror Formula:**
- The mirror formula is given by:
\[ \frac{1}{F} = \frac{1}{B} + \frac{1}{U} \]
- Substituting the values of B and U:
\[ \frac{1}{F} = \frac{1}{0.125} + \frac{1}{-0.5} \]

5. **Calculate the Values:**
- Calculate \( \frac{1}{B} \):
\[ \frac{1}{0.125} = 8 \]
- Calculate \( \frac{1}{U} \):
\[ \frac{1}{-0.5} = -2 \]
- Now substitute these values into the mirror formula:
\[ \frac{1}{F} = 8 - 2 = 6 \]

6. **Find the Focal Length (F):**
- Taking the reciprocal gives:
\[ F = \frac{1}{6} \approx 0.1667 \text{ m} \]
- Since it is a convex mirror, the focal length is positive:
\[ F \approx 0.167 \text{ m} \]

### Final Answer:
The focal length of the convex mirror is approximately **0.167 m**.