For the same statement as above, the ration of the two image sizes for these two positions of the lens is

Options

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

Explanation

To solve the problem of finding the ratio of the sizes of two images formed by a convex lens at two different positions, we can follow these steps:

### Step 1: Define the Variables
Let:
- \( D \) = initial distance from the object to the lens
- \( d \) = distance by which the lens is shifted
- \( u_1 \) = object distance for the first position of the lens
- \( u_2 \) = object distance for the second position of the lens
- \( v_1 \) = image distance for the first position of the lens
- \( v_2 \) = image distance for the second position of the lens

### Step 2: Calculate Object Distances
For the first position of the lens:
\[ u_1 = \frac{D - d}{2} \]

For the second position of the lens:
\[ u_2 = \frac{D + d}{2} \]

### Step 3: Calculate Image Distances
Using the lens formula, we can derive the image distances:

For the first position:
\[ v_1 = D - u_1 = D - \frac{D - d}{2} = \frac{D + d}{2} \]

For the second position:
\[ v_2 = D - u_2 = D - \frac{D + d}{2} = \frac{D - d}{2} \]

### Step 4: Calculate Magnifications
The magnification \( m \) is given by the formula:
\[ m = \frac{v}{u} \]

For the first position:
\[ m_1 = \frac{v_1}{u_1} = \frac{\frac{D + d}{2}}{\frac{D - d}{2}} = \frac{D + d}{D - d} \]

For the second position:
\[ m_2 = \frac{v_2}{u_2} = \frac{\frac{D - d}{2}}{\frac{D + d}{2}} = \frac{D - d}{D + d} \]

### Step 5: Calculate the Ratio of the Image Sizes
To find the ratio of the heights of the images formed at the two positions, we can use the magnification values:
\[ \text{Ratio} = \frac{m_1}{m_2} = \frac{\frac{D + d}{D - d}}{\frac{D - d}{D + d}} = \frac{(D + d)^2}{(D - d)^2} \]

### Final Answer
The ratio of the two image sizes for these two positions of the lens is:
\[ \frac{(D + d)^2}{(D - d)^2} \]