A lens is placed between the source of light and a wall. It forms images of area `A_(1)` and `A_(2)` on the wall for its two displaced positions. The area of source of light is

Options

A. `(A_(1)+A_(2))/(2)`
B. `sqrt((A_(1)A_(2))`
C. `[(1)/(A_(1))+(1)/(A_(2))]`
D. `((sqrt(A_(1))+sqrt(A_(2)))/(2))^(2)`

Answer

To solve the problem, we need to find the area of the source of light based on the areas of the images formed on the wall by the lens in two different positions. Let's denote the area of the source of light as \( A \).

### Step-by-Step Solution:

1. **Understanding the relationship between object and image areas**:
The area of the image formed by a lens is related to the area of the object (source of light) by the magnification (m). The magnification is defined as the ratio of the height of the image to the height of the object.

2. **Define magnification for both positions**:
Let \( m_1 \) be the magnification for the first position of the lens, and \( m_2 \) for the second position. The areas of the images formed in these two positions are \( A_1 \) and \( A_2 \) respectively.

The magnification can be expressed as:
\[ m_1 = \frac{A_1}{A} \quad \text{and} \quad m_2 = \frac{A_2}{A} \]

3. **Relating the areas**:
Since the magnification is also related to the height of the images and the object, we can express the areas of the images in terms of the area of the source:
\[ A_1 = m_1^2 \cdot A \quad \text{and} \quad A_2 = m_2^2 \cdot A \]

4. **Combining the equations**:
From the two equations above, we can express the area of the source \( A \) in terms of \( A_1 \) and \( A_2 \):
\[ A = \frac{A_1}{m_1^2} = \frac{A_2}{m_2^2} \]

5. **Finding the area of the source**:
Since both expressions equal \( A \), we can equate them:
\[ \frac{A_1}{m_1^2} = \frac{A_2}{m_2^2} \]

6. **Expressing the area of the source in terms of image areas**:
We can rearrange the equation to find the area of the source:
\[ A = \sqrt{A_1 \cdot A_2} \]

### Final Result:
The area of the source of light is given by:
\[ A = \sqrt{A_1 \cdot A_2} \]