In a convex mirror, of an object kept at a distance of 25 cm from mirror and half of its original height, focus of the mirror is

Options

A. 20 cm
B. 15 cm
C. 25 cm
D. 10 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
- Distance of the object from the mirror (u) = -25 cm (the negative sign is used because the object is in front of the mirror)
- The height of the image is half of the height of the object, which gives us the magnification (m) = 1/2.

### Step 2: Use the magnification formula
The magnification (m) is given by the formula:
\[ m = -\frac{v}{u} \]
Where:
- \( v \) = image distance from the mirror
- \( u \) = object distance from the mirror

Substituting the known values into the magnification formula:
\[ \frac{1}{2} = -\frac{v}{-25} \]

### Step 3: Solve for v
From the equation:
\[ \frac{1}{2} = \frac{v}{25} \]
Cross-multiplying gives:
\[ v = \frac{25}{2} = 12.5 \text{ cm} \]

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

Substituting the values of \( v \) and \( u \):
\[ \frac{1}{f} = \frac{1}{12.5} + \frac{1}{-25} \]

### Step 5: Calculate the individual terms
Calculating \( \frac{1}{12.5} \):
\[ \frac{1}{12.5} = \frac{2}{25} \]

Calculating \( \frac{1}{-25} \):
\[ \frac{1}{-25} = -\frac{1}{25} \]

### Step 6: Combine the fractions
Now substituting back into the mirror formula:
\[ \frac{1}{f} = \frac{2}{25} - \frac{1}{25} = \frac{1}{25} \]

### Step 7: Solve for f
Taking the reciprocal gives:
\[ f = 25 \text{ cm} \]

### Conclusion
The focal length of the convex mirror is \( 25 \text{ cm} \).

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