A luminous object and a screen are at a fixed distance D apart. A converging lens of focal length f is placed between the object and screen. A real image of the object in formed on the screen for two lens positins if they are separated by a distance d equal to

Options

A. `sqrt(D(D+4F))`
B. `sqrt(D(D-4F))`
C. `sqrt(2D(D-4F))`
D. `sqrt(D^(2)+4F)`

Explanation

To solve the problem, we will use the lens formula and the concept of image formation by a lens. Let's break down the solution step by step.

### Step 1: Understand the Setup
We have a luminous object and a screen fixed at a distance \( D \) apart. A converging lens of focal length \( f \) is placed between the object and the screen. We need to find the distance \( d \) between two positions of the lens where a real image is formed on the screen.

### Step 2: Use the Lens Formula
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
where:
- \( f \) = focal length of the lens
- \( u \) = object distance from the lens
- \( v \) = image distance from the lens

### Step 3: Set Up the Equations for Two Positions of the Lens
Let’s denote the object distance and image distance for the first position of the lens as \( u_1 \) and \( v_1 \), respectively. For the second position, let them be \( u_2 \) and \( v_2 \).

Since the object and screen are fixed at a distance \( D \):
\[ v_1 = D - u_1 \quad \text{(for the first position)} \]
\[ v_2 = D - u_2 \quad \text{(for the second position)} \]

### Step 4: Apply the Lens Formula for Both Positions
For the first position:
\[ \frac{1}{f} = \frac{1}{D - u_1} - \frac{1}{u_1} \]
Rearranging gives:
\[ \frac{1}{D - u_1} = \frac{1}{f} + \frac{1}{u_1} \]
Multiplying through by \( u_1(D - u_1)f \) leads to:
\[ u_1(D - u_1) = fu_1 + f(D - u_1) \]
This simplifies to:
\[ u_1D - u_1^2 = fu_1 + fD - fu_1 \]
\[ u_1D - u_1^2 = fD \]
\[ u_1^2 - u_1D + fD = 0 \quad \text{(1)} \]

For the second position:
\[ \frac{1}{f} = \frac{1}{D - u_2} - \frac{1}{u_2} \]
Following similar steps, we get:
\[ u_2^2 - u_2D + fD = 0 \quad \text{(2)} \]

### Step 5: Solve the Quadratic Equations
From equations (1) and (2), we can find the roots \( u_1 \) and \( u_2 \) using the quadratic formula:
\[ u = \frac{D \pm \sqrt{D^2 - 4fD}}{2} \]

### Step 6: Calculate the Distance \( d \)
The distance \( d \) between the two lens positions is given by:
\[ d = |u_2 - u_1| \]
Calculating \( u_2 - u_1 \):
\[ u_2 - u_1 = \frac{D + \sqrt{D^2 - 4fD}}{2} - \frac{D - \sqrt{D^2 - 4fD}}{2} \]
This simplifies to:
\[ u_2 - u_1 = \sqrt{D^2 - 4fD} \]

Thus, the distance \( d \) is:
\[ d = \sqrt{D^2 - 4fD} \]

### Conclusion
The distance \( d \) between the two positions of the lens where a real image is formed on the screen is:
\[ d = \sqrt{D^2 - 4fD} \]