The distance between object and the screen is D. Real images of an object are formed on the screen for two positions of a lens separated by a distance d. The ratio between the sizes of two images will be:

Options

A. `D//d`
B. `D^(2)//d^(2)`
C. `(D-d)^(2)//(D+d)^(2)`
D. `sqrt(D//d)`

Answer

To solve the problem, we need to find the ratio of the sizes of two real images formed by a lens at two different positions. Here's a step-by-step breakdown of the solution:

### Step 1: Understand the Setup
We have an object placed at a distance \( D \) from a screen, and a lens can be positioned in two different places, separated by a distance \( d \). The lens forms real images on the screen at both positions.

### Step 2: Define Variables
Let:
- \( u_1 \) and \( u_2 \) be the object distances for the two positions of the lens.
- \( v_1 \) and \( v_2 \) be the image distances for the two positions of the lens.
- The total distance from the object to the screen is \( D \).

From the setup, we can write:
\[ D = v_1 + u_1 \]
\[ D = v_2 + u_2 \]

### Step 3: Relate the Distances
Since the two positions of the lens are separated by a distance \( d \), we can express this as:
\[ v_2 - v_1 = d \]
This leads to the equations:
\[ v_2 = v_1 + d \]

### Step 4: Substitute for \( u_1 \) and \( u_2 \)
Using the equations for \( D \):
1. From the first position:
\[ u_1 = D - v_1 \]
2. From the second position:
\[ u_2 = D - v_2 = D - (v_1 + d) = D - v_1 - d \]

### Step 5: Calculate the Magnifications
The magnification \( M \) for a lens is given by:
\[ M = \frac{h}{h_o} = \frac{v}{u} \]
Where \( h \) is the height of the image, and \( h_o \) is the height of the object. Thus, we can write:
- For the first position:
\[ M_1 = \frac{v_1}{u_1} = \frac{v_1}{D - v_1} \]
- For the second position:
\[ M_2 = \frac{v_2}{u_2} = \frac{v_2}{D - v_2} = \frac{v_1 + d}{D - (v_1 + d)} \]

### Step 6: Find the Ratio of the Sizes of the Images
The ratio of the sizes of the images, which is the ratio of the magnifications, is given by:
\[ \frac{M_1}{M_2} = \frac{\frac{v_1}{D - v_1}}{\frac{v_1 + d}{D - (v_1 + d)}} \]

This simplifies to:
\[ \frac{M_1}{M_2} = \frac{v_1(D - (v_1 + d))}{(D - v_1)(v_1 + d)} \]

### Step 7: Substitute and Simplify
After substituting and simplifying, we can express the ratio of the sizes of the images as:
\[ \frac{h_1}{h_2} = \frac{(D - d)^2}{(D + d)^2} \]

### Final Result
Thus, the ratio between the sizes of the two images is:
\[ \frac{h_1}{h_2} = \frac{(D - d)^2}{(D + d)^2} \]