For a convex lens to form a real image, the distance between the object and the screen will be:

The question is about the distance between the object and the screen when a convex lens forms a real image. The usual textbook result is that this distance must be at least four times the focal length. The options don't say "at least," so we need to read the wording carefully.

Step 1: Analyze the question We need the condition on the object-to-screen distance D for a real image to form on a screen.

Step 2: The concept Use the lens formula:

1/f = 1/u + 1/v

Here u is the object distance and v is the image distance. Both are positive for a real object and a real image. The object-to-screen distance is D = u + v.

Step 3: Derivation From the lens formula, v = uf / (u − f).

So D = u + uf/(u − f) = u² / (u − f).

Rearranging gives a quadratic in u:

u² − Du + Df = 0

For u to be real (so the lens can be placed somewhere to form a sharp image), the discriminant must be non-negative:

D² − 4Df ≥ 0 D(D − 4f) ≥ 0 D ≥ 4f

Step 4: Interpret

• If D < 4f, no real image can form on the screen.
• If D = 4f, there is exactly one position, with u = v = 2f. The image is the same size as the object.
• If D > 4f, there are two possible lens positions.

So A, B and C are all impossible. A and B give a distance less than 4f, and C gives 2f, which is also less than 4f. Only D, four times the focal length, matches the condition. It is the minimum distance, and strictly it means "at least four times."

Answer: D. four times the focal length of the lens.

Tip: Remember "minimum object-to-screen distance = 4f." This is the case where u = v = 2f, so the image is real, inverted and the same size as the object. Many exam questions on this topic are really asking for this minimum.