A point object is placed on the axis of the concave mirror at a distance of 60 cm from the focal point of the mirror. Its image is formed at the point of object then focal length of the mirror is

Options

A. 15 cm
B. 30 cm
C. 60 cm
D. 120 cm

Answer

To solve the problem, we need to find the focal length of the concave mirror given that a point object is placed at a distance of 60 cm from the focal point and that its image is formed at the point of the object.

### Step-by-Step Solution:

1. **Understand the Given Information:**
- The object is placed at a distance of 60 cm from the focal point of the concave mirror.
- Let the focal length of the mirror be \( f \).
- The distance of the object from the mirror \( u \) can be expressed as:
\[ u = f + 60 \, \text{cm} \]
- The image is formed at the same point as the object, which means the image distance \( v \) is equal to \( u \):
\[ v = -u \]
(The negative sign indicates that the image is formed on the same side as the object for a concave mirror.)

2. **Use the Mirror Formula:**
- The mirror formula relates the object distance \( u \), the image distance \( v \), and the focal length \( f \):
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]

3. **Substituting Values:**
- Since \( v = -u \), we can substitute \( v \) in the mirror formula:
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{-u} = \frac{1}{u} - \frac{1}{u} = 0 \]
- This indicates that the object is at the center of curvature, which is twice the focal length.

4. **Finding the Focal Length:**
- Since \( u = f + 60 \), and we know that at the center of curvature \( u = 2f \):
\[ 2f = f + 60 \]
- Rearranging gives:
\[ 2f - f = 60 \implies f = 60 \, \text{cm} \]

### Final Answer:
The focal length of the concave mirror is \( 60 \, \text{cm} \).