A boy of height 1m stands in front of a convex mirror. His distance from the mirror is equal to its focal length. The height of his image is

Options

A. `0.25 m`
B. `0.33 m`
C. `0.5 m`
D. `0.67 m`

Explanation

To solve the problem step by step, we will use the concepts of ray optics, specifically the properties of convex mirrors.

### Step 1: Identify the given information
- Height of the boy (object height, \( h_o \)) = 1 m
- Distance of the boy from the mirror (object distance, \( u \)) = focal length of the convex mirror (let's denote it as \( f \))
- Since the object is in front of the mirror, we take \( u = -f \) (the negative sign indicates that the object is in front of the mirror).

**Hint:** Remember that in optics, distances measured in the direction of the incident light are negative.

### Step 2: Use the mirror formula
The mirror formula for a convex mirror is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Where:
- \( f \) = focal length of the mirror
- \( v \) = image distance
- \( u \) = object distance

Substituting \( u = -f \) into the mirror formula:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{f} \]

### Step 3: Rearranging the formula
Rearranging the equation to find \( \frac{1}{v} \):
\[ \frac{1}{v} = \frac{1}{f} + \frac{1}{f} = \frac{2}{f} \]
Thus, we can find \( v \):
\[ v = \frac{f}{2} \]

**Hint:** Make sure to keep track of the signs when substituting values into the mirror formula.

### Step 4: Calculate the magnification
The magnification \( m \) of a mirror is given by the formula:
\[ m = -\frac{h_i}{h_o} = \frac{v}{u} \]
Substituting the values we have:
\[ m = \frac{v}{u} = \frac{\frac{f}{2}}{-f} = -\frac{1}{2} \]

### Step 5: Find the height of the image
Using the magnification formula:
\[ m = -\frac{h_i}{h_o} \]
We can rearrange it to find the height of the image \( h_i \):
\[ h_i = -m \cdot h_o \]
Substituting \( m = -\frac{1}{2} \) and \( h_o = 1 \, \text{m} \):
\[ h_i = -\left(-\frac{1}{2}\right) \cdot 1 = \frac{1}{2} \, \text{m} \]

### Final Answer
The height of the image is \( 0.5 \, \text{m} \).

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