A lense of focal length f is placed in between an object and screen at a distance D. The lens forms two real images of object on the screen for two of its different positions, a distance x apart. The two real images have magnification `m_1 and m_2` respectively `(m_1gtm_2).` Choose the correct statement(s).

Options

A. `m_1m_2=-1`
B. `m_1m_2=1`
C. `f=(D^2-x^2)/(4D)`
D. `Dge4f`

Explanation

To solve the problem, we need to analyze the situation involving a lens, an object, and a screen. The lens forms two real images of the object on the screen when the lens is moved slightly. We will derive the necessary relationships and identify the correct statements based on the given conditions.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a lens with focal length \( f \).
- The distance between the lens and the screen is \( D \).
- The lens is moved slightly, creating two positions that yield two different real images on the screen, which are \( x \) apart.

2. **Magnification Relationship**:
- The magnification \( m \) of a lens is given by the formula:
\[ m = \frac{h'}{h} = \frac{v}{u} \]
where \( h' \) is the height of the image, \( h \) is the height of the object, \( v \) is the image distance, and \( u \) is the object distance.
- For the two positions of the lens, we denote the magnifications as \( m_1 \) and \( m_2 \).

3. **Using the Displacement Method**:
- According to the displacement method for lenses, when the lens is moved, the product of the magnifications of the two images formed is equal to 1:
\[ m_1 \cdot m_2 = 1 \]
- This indicates that if one image is larger (greater magnification), the other must be smaller (lesser magnification).

4. **Focal Length Relationship**:
- The focal length \( f \) can be expressed in terms of the distance \( D \) and the distance \( x \) between the two images:
\[ f = \frac{D^2 - x^2}{4D} \]
- This formula arises from the geometry of the lens and the positions of the images.

5. **Condition on Distance**:
- It is also given that \( D \) must be greater than or equal to \( 4f \):
\[ D \geq 4f \]
- This condition ensures that the lens can form real images at the specified distances.

6. **Conclusion**:
- From the above relationships, we can conclude that:
- The product of the magnifications \( m_1 \) and \( m_2 \) is equal to 1.
- The focal length can be calculated using the given formula.
- The distance \( D \) must satisfy the condition \( D \geq 4f \).

### Correct Statements:
Based on the analysis, the correct statements are:
- \( m_1 \cdot m_2 = 1 \)
- \( f = \frac{D^2 - x^2}{4D} \)
- \( D \geq 4f \)