For a concave lens of focal length f, the relation between object and image distance u and v, respectively, from its pole can best be represented by (u = v is the reference line) :

Options

A.
B.
C.
D.

Answer

To solve the problem regarding the relationship between object distance \( u \) and image distance \( v \) for a concave lens with focal length \( f \), we will follow these steps:

### Step-by-Step Solution:

1. **Understand the Sign Convention**:
- For a concave lens, the focal length \( f \) is considered negative.
- The object distance \( u \) is also taken as negative since the object is placed on the same side as the incoming light.
- The image distance \( v \) is negative as well since the image formed by a concave lens is virtual and located on the same side as the object.

2. **Lens Formula**:
- The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
- Rearranging this, we can express \( v \) in terms of \( u \):
\[ \frac{1}{v} = \frac{1}{f} - \frac{1}{u} \]
- This can be rewritten as:
\[ v = \frac{fu}{u + f} \]

3. **Graphical Representation**:
- To analyze the relationship graphically, we can plot \( v \) against \( u \).
- The equation \( v = \frac{fu}{u + f} \) indicates that as \( u \) approaches infinity (the object moves far away), \( v \) approaches \( f \) (which is negative).

4. **Finding the Tangent**:
- At the origin (where \( u = 0 \)), we can find the slope of the curve to determine the tangent.
- The derivative \( \frac{dv}{du} \) can be calculated to find the slope at \( u = 0 \):
\[ \frac{dv}{du} = \frac{f}{(u + f)^2} \]
- Evaluating this at \( u = 0 \):
\[ \frac{dv}{du} \bigg|_{u=0} = \frac{f}{f^2} = \frac{1}{f} \]

5. **Conclusion**:
- The tangent at the origin corresponds to the line \( v = u \) since it has a slope of 1.
- The curve will not intersect the line \( v = u \) except at the origin, confirming that the relationship between \( u \) and \( v \) for a concave lens does not allow for real intersections.

### Final Answer:
The correct representation of the relationship between object distance \( u \) and image distance \( v \) for a concave lens is that the curve approaches the line \( v = u \) but does not intersect it except at the origin.