Which of the following graph represents relation between the image position (v) and object position (u) from the concave lens.

Options

A. Option1
B. Option2
C. Option3
D. Option4

Answer

To solve the problem of determining which graph represents the relationship between the image position (v) and object position (u) for a concave lens, we will follow these steps:

### Step 1: Understand the Lens Formula
The lens formula is given by:
\[ \frac{1}{v} - \frac{1}{u} = \frac{1}{f} \]
where:
- \( v \) = image distance
- \( u \) = object distance
- \( f \) = focal length of the lens

### Step 2: Identify the Sign Convention
For a concave lens:
- The focal length \( f \) is negative.
- The object distance \( u \) is taken as negative when the object is placed in front of the lens (which is the usual case).

### Step 3: Rewrite the Lens Formula
Substituting \( f \) as negative in the lens formula, we get:
\[ \frac{1}{v} - \frac{1}{u} = -\frac{1}{|f|} \]
This can be rearranged to:
\[ \frac{1}{v} = \frac{1}{u} - \frac{1}{|f|} \]
This indicates that as \( u \) increases (becomes less negative), \( v \) will also change.

### Step 4: Analyze the Relationship
From the rearranged formula, we can see that:
- As \( u \) approaches \( -\infty \) (object far away), \( v \) approaches \( -f \) (image position approaches the focal point).
- As \( u \) approaches \( 0 \) (object at the focal point), \( v \) approaches \( -\infty \) (image moves further away).

### Step 5: Graphical Representation
To plot the graph:
- The x-axis represents the object position \( u \) (negative values).
- The y-axis represents the image position \( v \) (also negative values).
- The graph will start from the point where \( u \) is at \( -\infty \) and \( v \) approaches \( -f \).
- As \( u \) increases towards \( 0 \), \( v \) will drop towards \( -\infty \).

### Conclusion
The graph representing the relationship between image position \( v \) and object position \( u \) for a concave lens will be a hyperbola that approaches the x-axis and y-axis but never touches them, indicating that both \( v \) and \( u \) are negative.

### Final Answer
The correct option is the one that shows a hyperbolic curve representing this relationship.