A student measures the focal length of a convex lens by putting an object pin at a distance `u` from the lens and measuring the distance `v` of the image pin. The graph between `u` and `v` plotted by the student should look like

Options

A.
B.
C.
D.

Explanation

To solve the problem of how the graph between the object distance \( u \) and the image distance \( v \) for a convex lens looks, we can follow these steps:

### Step 1: Understand the Lens Formula
The lens formula for a convex lens is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
where:
- \( f \) is the focal length of the lens (positive for a convex lens),
- \( u \) is the object distance (negative in the Cartesian convention),
- \( v \) is the image distance (positive for real images).

### Step 2: Rearranging the Lens Formula
We can rearrange the lens formula to express \( v \) in terms of \( u \):
\[ \frac{1}{v} = \frac{1}{f} + \frac{1}{u} \]
Taking the reciprocal gives:
\[ v = \frac{fu}{u + f} \]

### Step 3: Analyze the Relationship
From the rearranged formula, we can see that as \( u \) increases (moving the object further away from the lens), \( v \) will also increase. However, the relationship is not linear; it is hyperbolic because of the \( \frac{1}{u} \) term.

### Step 4: Determine the Nature of the Graph
Since the relationship between \( u \) and \( v \) is of the form \( v = \frac{fu}{u + f} \), it indicates a hyperbolic relationship. The graph will curve towards the axes, specifically in the first quadrant where \( u \) is negative and \( v \) is positive.

### Step 5: Identify the Correct Graph Option
Given the nature of the graph (hyperbolic), we can eliminate options that suggest a linear relationship. The correct graph should show a curve that approaches the axes but never touches them, confirming that as \( u \) approaches negative infinity, \( v \) approaches \( f \).

### Conclusion
The graph between \( u \) and \( v \) plotted by the student should look like a hyperbola in the first quadrant, confirming that the correct option is option C.

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