A point object is placed at distance of 30 cm in front of a convex mirror of Focal length 30 cm. The image will form at

Options

A. infinity
B. pole
C. focus
D. 15 cm behind the mirror

Answer

To find the position of the image formed by a convex mirror when a point object is placed in front of it, we can use the mirror formula:

**Mirror Formula:**
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]

Where:
- \( f \) = focal length of the mirror
- \( v \) = image distance from the mirror
- \( u \) = object distance from the mirror

### Step 1: Identify the given values
- The focal length \( f \) of the convex mirror is given as \( 30 \, \text{cm} \). Since it is a convex mirror, we take \( f \) as positive:
\[ f = +30 \, \text{cm} \]
- The object distance \( u \) is given as \( 30 \, \text{cm} \) in front of the mirror. According to the sign convention, we take \( u \) as negative:
\[ u = -30 \, \text{cm} \]

### Step 2: Substitute the values into the mirror formula
Using the mirror formula:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the known values:
\[ \frac{1}{30} = \frac{1}{v} + \frac{1}{-30} \]

### Step 3: Rearranging the equation
To isolate \( \frac{1}{v} \), we rearrange the equation:
\[ \frac{1}{v} = \frac{1}{30} + \frac{1}{30} \]
\[ \frac{1}{v} = \frac{1 + (-1)}{30} = \frac{1 + 1}{30} = \frac{2}{30} \]
\[ \frac{1}{v} = \frac{1}{15} \]

### Step 4: Solve for \( v \)
Taking the reciprocal to find \( v \):
\[ v = 15 \, \text{cm} \]

### Step 5: Determine the position of the image
Since \( v \) is positive, it indicates that the image is formed on the same side as the object, but behind the mirror. Thus, the image is located \( 15 \, \text{cm} \) behind the convex mirror.

### Final Answer:
The image will be formed at a distance of \( 15 \, \text{cm} \) behind the mirror.

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