Introduction:
When a convex lens is placed between an object and a screen, two real images of the object are formed for two different positions of the lens. In this proof, we aim to show that the length of the object (L) is equal to the square root of the product of the lengths of the two real images (L1 and L2).

Proof:

Step 1: Image Formation:
When a convex lens is placed between an object and a screen, the lens refracts the light rays from the object and forms an image on the screen. The image can be either real or virtual, depending on the position of the object and the lens.

Step 2: Position of the Lens:
Let's consider two different positions of the convex lens, denoted as L1 and L2. In both positions, the lens forms real images of the object on the screen.

Step 3: Length of the Object:
The length of the object is denoted as L. It represents the physical dimension of the object along the principal axis.

Step 4: Length of the Images:
The lengths of the two real images formed by the lens in positions L1 and L2 are denoted as L1 and L2, respectively. These lengths represent the physical dimensions of the images along the principal axis.

Step 5: Lens Formula:
According to the lens formula, 1/f = 1/v - 1/u, where f is the focal length of the lens, v is the image distance, and u is the object distance. Since the lens forms real images in both positions (L1 and L2), the image distances (v1 and v2) will be positive.

Step 6: Magnification:
The magnification (m) of the lens is given by the formula m = -v/u, where m represents the size of the image with respect to the object. As the lens forms real images, the magnification will be negative.

Step 7: Using Lens Formula and Magnification:
For position L1: 1/f = 1/v1 - 1/u1 and m1 = -v1/u1
For position L2: 1/f = 1/v2 - 1/u2 and m2 = -v2/u2

Step 8: Equating the Magnifications:
Since the magnification is negative for both positions, we can equate the magnitudes: |m1| = |m2|, which implies that (v1/u1) = (v2/u2)

Step 9: Using the Lens Formula:
From the lens formula, we can rewrite the equations as:
1/f = 1/v1 - 1/u1 = 1/v2 - 1/u2

Step 10: Applying the Mirror Formula:
We can use the mirror formula, 1/f = 1/v - 1/u, to determine the relationship between the object distance (u) and the image distance (v) for both positions L1 and L2.

Step 11: Relationship between L1 and L2:
From the mirror formula, we have:
1/f = 1/v1 - 1/u1