Proof:

Given:
- A convex lens is placed between an object and a screen.
- Two real images of the object are formed for two positions of the lens.
- Let L1 and L2 be the lengths of the two real images.
- Let L be the length of the object.

To prove:
L = √(L1 * L2)

Proof:

Step 1: Understanding the Image Formation by a Convex Lens

When an object is placed in front of a convex lens, it forms an image on the other side of the lens. The image can be either real or virtual, depending on the position of the object with respect to the lens.

In this case, two real images are formed for two positions of the lens. This means that the object is placed beyond the focal point of the lens in both cases.

Step 2: Relationship between Object Distance, Image Distance, and Focal Length

For a convex lens, the relationship between the object distance (u), image distance (v), and focal length (f) is given by the lens formula:

1/f = 1/v - 1/u

In this case, since the object is placed beyond the focal point, the image distance (v) is positive.

Step 3: Calculating the Length of the Object and the Real Images

The length of the object (L) can be calculated using the formula:

L = h/u

where h is the height of the object and u is the object distance.

Similarly, the lengths of the real images (L1 and L2) can be calculated using the formula:

L1 = h1/v1
L2 = h2/v2

where h1 and h2 are the heights of the real images, and v1 and v2 are the image distances.

Step 4: Relating the Object Length and the Real Image Lengths

To prove that L = √(L1 * L2), we need to relate the object length (L) to the lengths of the real images (L1 and L2).

From Step 3, we have:

L1 = h1/v1
L2 = h2/v2

Cross-multiplying these equations, we get:

h1 = L1 * v1
h2 = L2 * v2

Now, let's calculate the product of the lengths of the real images:

L1 * L2 = (h1/v1) * (h2/v2)

Multiplying the numerators and denominators, we get:

L1 * L2 = (h1 * h2) / (v1 * v2)

Since h1 * h2 is equal to the square of the height of the object (h^2), we can rewrite the equation as:

L1 * L2 = (h^2) / (v1 * v2)

Using the lens formula (1/f = 1/v - 1/u), we can write:

1/v1 = 1/f - 1/u1
1/v2 = 1/f - 1/u2

Substituting these values into the equation, we get:

L1 * L2 = (h^2) / [(1/f - 1/u1) * (1