An iron nail is kept at a distance of 50 cm in front of a convex mirror. The height of the image formed is half of the height of object. Where should we place the pin to get an image of one-fourth the height of object ?

Explanation

To solve the problem step by step, we will follow the principles of optics related to convex mirrors and magnification.

### Step 1: Understand the given data
- The object (iron nail) is placed at a distance \( u = -50 \, \text{cm} \) (the negative sign indicates that the object is in front of the mirror).
- The height of the image is half the height of the object, which means the magnification \( m = \frac{h_i}{h_o} = \frac{1}{2} \).

### Step 2: Relate magnification to image distance
For a convex mirror, the magnification can also be expressed as:
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