A point object is placed at a distance of 30 cm from a convex mirror of focal length 30 cm. The image will form at

Options

A. Infinity
B. Focus
C. Pole
D. 15 cm behind the mirror

Answer

To solve the problem of finding the image distance formed by a convex mirror when a point object is placed at a distance of 30 cm, we will use the mirror formula:

\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]

where:
- \( f \) is the focal length of the mirror,
- \( u \) is the object distance (taken as negative for real objects),
- \( v \) is the image distance (which we need to find).

### Step-by-step Solution:

1. **Identify the Given Values:**
- Focal length of the convex mirror, \( f = +30 \) cm (positive for convex mirrors).
- Object distance, \( u = -30 \) cm (negative because the object is in front of the mirror).

2. **Apply the Mirror Formula:**
Substitute the values into the mirror formula:
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]
This becomes:
\[ \frac{1}{30} = \frac{1}{-30} + \frac{1}{v} \]

3. **Rearranging the Equation:**
To isolate \( \frac{1}{v} \), we can rearrange the equation:
\[ \frac{1}{v} = \frac{1}{30} + \frac{1}{30} \]

4. **Combine the Fractions:**
\[ \frac{1}{v} = \frac{1 + 1}{30} = \frac{2}{30} \]
Simplifying this gives:
\[ \frac{1}{v} = \frac{1}{15} \]

5. **Find the Image Distance \( v \):**
Taking the reciprocal of both sides:
\[ v = 15 \text{ cm} \]

6. **Determine the Nature of the Image:**
Since \( v \) is positive, this indicates that the image is virtual and located behind the mirror.

### Conclusion:
The image will form at a distance of 15 cm behind the convex mirror.