In an optics experiment, with the position of the object fixed, a student varies the position of a convex lens and for each position, the screen is adjusted to get a clear image of the object. A graph between the object distance u and the image distance v, from the lens, is plitted using the same scale for the two axes. A straight line passing through the origin and making an angle of `45^circ` with x-axis meets the experimental curve at P. The coordinates of P will be.

Options

A. `((f)/(2),(f)/(2))`
B. `(f,f)`
C. `(4f,4f)`
D. `(2f,2f)

Explanation

To solve the problem, we need to analyze the relationship between the object distance (u) and the image distance (v) for a convex lens using the lens formula and the information provided about the graph.

### Step-by-Step Solution:

1. **Understanding the Lens Formula**:
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
where \( f \) is the focal length of the lens, \( u \) is the object distance, and \( v \) is the image distance.

2. **Graphing the Relationship**:
When we plot the graph of \( u \) (object distance) on the x-axis and \( v \) (image distance) on the y-axis, we can rearrange the lens formula to find the relationship between \( u \) and \( v \):
\[ \frac{1}{v} = \frac{1}{f} + \frac{1}{u} \]
This represents a hyperbolic relationship between \( u \) and \( v \).

3. **Identifying the 45-degree Line**:
The problem states that there is a straight line passing through the origin making an angle of \( 45^\circ \) with the x-axis. The equation of this line can be expressed as:
\[ v = u \]
This means that at the point of intersection \( P \), the object distance \( u \) is equal to the image distance \( v \).

4. **Finding the Coordinates of Point P**:
Since the line \( v = u \) intersects the hyperbola formed by the lens formula, we can substitute \( v \) with \( u \) in the lens formula:
\[ \frac{1}{f} = \frac{1}{u} - \frac{1}{u} \]
This simplifies to:
\[ \frac{1}{f} = 0 \]
This indicates that the point \( P \) corresponds to the condition where the object is at the center of curvature of the lens.

5. **Determining the Center of Curvature**:
The center of curvature \( C \) for a convex lens is located at a distance of \( 2f \) from the lens. Therefore, the coordinates of point \( P \) where \( u = v \) will be:
\[ P(2f, 2f) \]

### Final Answer:
The coordinates of point \( P \) are \( (2f, 2f) \).