An object is placed at 50 cm in front of a concave mirror of focal length 25 cm. What is the nature of the image produced by the mirror ?

Options

A. real and erect and of the same size
B. virtual and inverted and bigger in size
C. real, inverted and of the same size
D. real, inverted and smaller in size

Explanation

To determine the nature of the image produced by a concave mirror when an object is placed in front of it, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Object distance (u) = -50 cm (the negative sign indicates that the object is in front of the mirror)
- Focal length (f) = -25 cm (the negative sign indicates that it is a concave mirror)

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

3. **Substitute the Values into the Mirror Formula:**
\[ \frac{1}{-25} = \frac{1}{v} + \frac{1}{-50} \]

4. **Rearranging the Equation:**
\[ \frac{1}{v} = \frac{1}{-25} + \frac{1}{50} \]

5. **Finding a Common Denominator:**
The common denominator for -25 and 50 is 50:
\[ \frac{1}{v} = \frac{-2}{50} + \frac{1}{50} = \frac{-2 + 1}{50} = \frac{-1}{50} \]

6. **Calculate Image Distance (v):**
\[ v = -50 \text{ cm} \]
The negative sign indicates that the image is formed on the same side as the object.

7. **Determine the Nature of the Image:**
- Since the image distance (v) is negative, the image is real.
- The image is inverted because it is formed on the same side as the object.
- The image size can be determined by the magnification formula, but since the object is at the center of curvature, the image will be of the same size.

### Conclusion:
The nature of the image produced by the concave mirror is:
- **Real**
- **Inverted**
- **Same size as the object**