The image formed by a convex mirror of focal length `30 cm`. is a quarter of the object. What is the distance of the object from the mirror ?

Options

A. 30 cm
B. 90 cm
C. 120 cm
D. 60 cm

Explanation

To solve the problem, we need to find the distance of the object from a convex mirror given that the focal length is 30 cm and the image formed is a quarter of the object.

### Step-by-Step Solution:

1. **Identify the given values**:
- Focal length of the convex mirror, \( f = +30 \, \text{cm} \) (positive because it is a convex mirror).
- Magnification, \( m = \frac{1}{4} \).

2. **Use the magnification formula**:
The magnification \( m \) for a mirror is given by the formula:
\[ m = \frac{h'}{h} = -\frac{v}{u} \]
where \( h' \) is the height of the image, \( h \) is the height of the object, \( v \) is the image distance, and \( u \) is the object distance. Since the image is virtual and upright in a convex mirror, we can use the positive value for magnification:
\[ m = \frac{1}{4} = -\frac{v}{u} \]

3. **Rearranging the magnification formula**:
From the magnification formula, we can express \( v \) in terms of \( u \):
\[ v = -\frac{1}{4} u \]

4. **Use the mirror formula**:
The mirror formula relates the focal length \( f \), object distance \( u \), and image distance \( v \):
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the known values:
\[ \frac{1}{30} = \frac{1}{-\frac{1}{4}u} + \frac{1}{u} \]

5. **Finding a common denominator**:
The common denominator for the right side is \( -\frac{1}{4}u \cdot u = -\frac{u^2}{4} \):
\[ \frac{1}{30} = -\frac{4}{u} + \frac{4}{u} = \frac{-4 + 4}{u} = \frac{4}{-u} \]

6. **Solving for \( u \)**:
Rearranging gives:
\[ \frac{1}{30} = \frac{4}{-u} \]
Cross-multiplying:
\[ -u = 120 \implies u = -120 \, \text{cm} \]

7. **Conclusion**:
The object distance from the mirror is \( 120 \, \text{cm} \) (the negative sign indicates that the object is in front of the mirror).

### Final Answer:
The distance of the object from the mirror is \( 120 \, \text{cm} \).