An object is placed in from of a convex mirror of focal length 50 cm, if the image is one fourth the size of the object, the distance of the image from the mirror is

Options

A. 25 cm
B. 37.5 cm
C. 50 cm
D. 75 cm

Explanation

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

### Step 1: Identify the given values
- Focal length (f) of the convex mirror = +50 cm (positive because it is a convex mirror)
- Magnification (m) = 1/4 (the image is one fourth the size of the object)

### Step 2: Set up the object distance (u)
- The object distance (u) is always negative for mirrors. Let’s denote it as:
\[ u = -x \quad \text{(where x is the magnitude of the object distance)} \]

### Step 3: Use the magnification formula
The magnification (m) is given by:
\[ m = -\frac{v}{u} \]
Substituting the values we have:
\[ \frac{1}{4} = -\frac{v}{-x} \implies v = \frac{x}{4} \]

### Step 4: Apply the mirror formula
The mirror formula is given by:
\[ \frac{1}{v} + \frac{1}{u} = \frac{1}{f} \]
Substituting the known values:
\[ \frac{1}{\frac{x}{4}} + \frac{1}{-x} = \frac{1}{50} \]
This simplifies to:
\[ \frac{4}{x} - \frac{1}{x} = \frac{1}{50} \]
Combining the fractions on the left side:
\[ \frac{4 - 1}{x} = \frac{1}{50} \implies \frac{3}{x} = \frac{1}{50} \]

### Step 5: Solve for x
Cross-multiplying gives:
\[ 3 \cdot 50 = x \implies x = 150 \text{ cm} \]
Thus, the object distance \( u = -150 \text{ cm} \).

### Step 6: Find the image distance (v)
Using the relation we found earlier:
\[ v = \frac{x}{4} = \frac{150}{4} = 37.5 \text{ cm} \]

### Conclusion
The distance of the image from the mirror is:
\[ v = 37.5 \text{ cm} \]