Rays of light from Sunn falls on a biconvex lens of focal length f an the circular image of Sun of radius r is formed on the focal plane of the lens. Then,

Options

A. area of image is `pir^(2)` and area is directly proportinal of f
B. area of image is `pir^(2)` and area is directly proportional to `t^(2)`
C. intensity of image increases if f is increases
D. If lower half of the lens is covered with black paper, area will become half

Answer

To solve the problem, we need to analyze the information given about the biconvex lens and the image formed by it.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a biconvex lens with a focal length \( f \).
- Rays of light from the Sun (considered to be at infinity) converge to form a circular image of the Sun on the focal plane of the lens.
- The radius of this circular image is given as \( r \).

2. **Identifying the Image Properties**:
- The image formed is circular, and its radius is \( r \). Therefore, the diameter of the image is \( 2r \).

3. **Using Geometry**:
- The rays from the Sun are parallel when they reach the lens. After passing through the lens, they converge at the focal plane.
- The distance from the center of the lens to the edge of the image is \( r \), and the distance from the lens to the focal plane is \( f \).

4. **Applying Trigonometry**:
- We can use the tangent of the angle formed by the rays at the lens. Let \( \beta \) be the angle of incidence.
- From the geometry, we have:
\[ \tan(\beta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{r}{f} \]
- Therefore, we can express \( r \) in terms of \( f \) and \( \tan(\beta) \):
\[ r = f \tan(\beta) \]

5. **Finding the Area of the Image**:
- The area \( A \) of the circular image is given by:
\[ A = \pi r^2 \]
- Substituting \( r = f \tan(\beta) \) into the area formula gives:
\[ A = \pi (f \tan(\beta))^2 = \pi f^2 \tan^2(\beta) \]

6. **Analyzing the Proportionality**:
- From the area equation \( A = \pi f^2 \tan^2(\beta) \), we can see that the area \( A \) is directly proportional to \( f^2 \) (assuming \( \tan(\beta) \) is constant for small angles).

7. **Evaluating the Given Statements**:
- **Statement A**: The area of the image is \( \pi r^2 \) and is directly proportional to \( f \) - **Incorrect**.
- **Statement B**: The area of the image is \( \pi r^2 \) and is directly proportional to \( f^2 \) - **Correct**.
- **Statement C**: The intensity of the image increases if \( f \) increases - **Incorrect** (intensity depends on area and light intensity).
- **Statement D**: If the lower half of the lens is covered, the area will become half - **Incorrect** (the area will be half, but the intensity will change).

### Conclusion:
The correct statement is **Statement B**: The area of the image is \( \pi r^2 \) and is directly proportional to \( f^2 \).