A convex mirror of focal length 'f' is placed at the origin with its reflecting surface towards the negative x-axis . Choose the correct graps between 'v' and 'u' for u`lt 0`.

Options

A.
B.
C.
D.

Explanation

To solve the problem, we need to analyze the relationship between the object distance (u) and the image distance (v) for a convex mirror. The mirror formula we will use is:

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

Where:
- \( v \) is the image distance,
- \( u \) is the object distance (which is negative for real objects in this case),
- \( f \) is the focal length of the convex mirror (which is positive).

### Step-by-Step Solution:

1. **Understanding the Sign Convention**:
- For a convex mirror, the focal length \( f \) is positive.
- The object distance \( u \) is negative because the object is placed on the negative x-axis (to the left of the mirror).
- The image distance \( v \) will also be positive since the image formed by a convex mirror is virtual and located on the positive side of the axis.

2. **Using the Mirror Formula**:
- Rearranging the mirror formula:
\[ \frac{1}{v} = \frac{1}{f} - \frac{1}{u} \]
- Since \( u < 0 \), we can express it as:
\[ \frac{1}{v} = \frac{1}{f} + \frac{1}{|u|} \]
Here, \( |u| \) is the magnitude of \( u \).

3. **Analyzing the Relationship**:
- As \( |u| \) increases (meaning the object moves further away from the mirror), \( \frac{1}{|u|} \) decreases, which means \( \frac{1}{v} \) approaches \( \frac{1}{f} \).
- Therefore, \( v \) approaches \( f \) as \( |u| \) approaches infinity.

4. **Finding Specific Values**:
- If \( u \) approaches 0 (which is not physically possible for a real object but helps in understanding the trend), then:
\[ v \to 0 \]
- If \( u \) approaches \(-\infty\), then \( v \) approaches \( f \).

5. **Graphing the Relationship**:
- The graph of \( v \) versus \( u \) will show that as \( u \) becomes less negative (approaching 0), \( v \) also approaches 0.
- As \( u \) becomes more negative (approaching \(-\infty\)), \( v \) approaches \( f \).

### Conclusion:
The correct graph between \( v \) and \( u \) for \( u < 0 \) will show a curve that starts from \( v = 0 \) when \( u = 0 \) and approaches \( v = f \) as \( |u| \) increases.