The graph showing correctly the variation of image distance `(v)` as a function of object distance `(u)` in case of a concave mirror is

Options

A.
B.
C.
D.

Answer

To solve the problem of determining the correct graph that shows the variation of image distance \( v \) as a function of object distance \( u \) for a concave mirror, we can follow these steps:

### Step 1: Understand the Mirror Formula
The mirror formula for a concave mirror is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Where:
- \( f \) is the focal length,
- \( v \) is the image distance,
- \( u \) is the object distance.

### Step 2: Rearrange the Mirror Formula
We can rearrange the mirror formula to express \( v \) in terms of \( u \):
\[ \frac{1}{v} = \frac{1}{f} - \frac{1}{u} \]
This can be rewritten as:
\[ v = \frac{fu}{u - f} \]

### Step 3: Analyze the Behavior of \( v \)
Now, let's analyze the behavior of \( v \) as \( u \) changes:
- As \( u \) approaches infinity (\( u \to \infty \)):
\[ v \to f \quad \text{(the image distance approaches the focal length)} \]
- As \( u \) approaches the focal point (\( u \to f \)):
\[ v \to \infty \quad \text{(the image distance becomes very large)} \]
- As \( u \) approaches zero (\( u \to 0 \)):
\[ v \to -f \quad \text{(the image distance becomes negative, indicating a virtual image)} \]

### Step 4: Sketch the Graph
Based on the analysis:
- The graph should start from negative values (when \( u \) is close to 0) and increase towards the focal length \( f \) as \( u \) increases.
- The graph should show that as \( u \) increases from 0 to infinity, \( v \) transitions from negative to positive values, eventually approaching \( f \).

### Step 5: Identify the Correct Option
From the options provided, we need to identify which graph correctly represents this behavior:
- The correct graph should show that \( v \) starts from a negative value (indicating a virtual image) and increases towards \( f \) as \( u \) increases.

### Conclusion
Based on the analysis, the correct option is **Option A**, which accurately depicts the relationship between \( v \) and \( u \) for a concave mirror.

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