In an experiment to determine the focal length `(f)` of a concave mirror by the `u-v` method, a student places the object pin A on the principal axis at a distance x from the pole P. The student looks at the pin and its inverted image from a distance keeping his/her eye in line with PA. When the student shifts his/her eye towards left, the image appears to the right of the object pin. Then,

Options

A. `xltf`
B. `fltxlt2f`
C. `x=2f`
D. `xgt2f`

Explanation

To solve the problem regarding the focal length of a concave mirror using the u-v method, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- A concave mirror is used, and an object pin A is placed on the principal axis at a distance \( x \) from the pole \( P \) of the mirror.
- The student observes the object pin and its inverted image.

2. **Observation of Image Position**:
- When the student shifts their eye to the left, the inverted image appears to the right of the object pin.
- This indicates that the image formed by the concave mirror is on the same side as the object pin.

3. **Analyzing Image Formation**:
- For a concave mirror, when the object is placed beyond the focal point, the image formed is real and inverted.
- If the image appears to the right of the object pin when the student shifts their eye, it suggests that the object is located beyond the center of curvature (C) of the mirror.

4. **Applying the Mirror Formula**:
- The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]
where \( f \) is the focal length, \( u \) is the object distance (negative for concave mirrors), and \( v \) is the image distance (positive for real images).

5. **Determining Focal Length**:
- Since the image is real and inverted, we can conclude that \( v \) is positive and greater than \( u \) (since the image is formed on the same side as the object).
- By substituting the known values into the mirror formula, we can solve for \( f \).

6. **Conclusion**:
- The focal length \( f \) can be determined based on the distances \( u \) and \( v \) obtained from the setup.

### Final Answer:
The focal length \( f \) of the concave mirror can be calculated using the distances measured from the object pin and the image position.

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