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

Answer

To solve the problem, we need to analyze the relationship between the object distance (u) and the image distance (v) for a convex lens, and how they relate to the straight line that makes a 45-degree angle with the x-axis.

### Step-by-Step Solution:

1. **Understanding 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, \( v \) is the image distance, and \( u \) is the object distance.

2. **Setting Up the Graph**:
We are plotting a graph with object distance \( u \) on the x-axis (which is negative for real objects) and image distance \( v \) on the y-axis. The line that makes a 45-degree angle with the x-axis has a slope of 1, meaning:
\[ v = -u \]
(since \( u \) is negative, we take the absolute value).

3. **Finding the Intersection Point**:
The line \( v = -u \) intersects the experimental curve at point P. At this point, we have:
\[ |v| = |u| \]
This means that the magnitudes of \( u \) and \( v \) are equal.

4. **Substituting into the Lens Formula**:
Since \( |v| = |u| \), we can substitute \( v = -u \) into the lens formula:
\[ \frac{1}{f} = \frac{1}{-u} - \frac{1}{u} \]
This simplifies to:
\[ \frac{1}{f} = -\frac{1}{u} - \frac{1}{u} = -\frac{2}{u} \]
Rearranging gives:
\[ u = -2f \]
Since \( v = -u \), we also have:
\[ v = 2f \]

5. **Coordinates of Point P**:
The coordinates of point P, where the line intersects the curve, are:
\[ (u, v) = (-2f, 2f) \]

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