Geometrical optics — for a thin convex lens of focal length f, if f₁ and f₂ are the distances from the optical centre to two conjugate points P₁ and P₂ respectively, which relationship correctly relates f, f₁, and f₂?

Introduction / Context:
Conjugate points P₁ and P₂ for a thin lens are a pair of object–image points that map into each other through the lens. For paraxial rays, their distances f₁ and f₂ from the optical centre obey the thin-lens formula. Remembering the correct algebraic relationship (with proper sign convention) is vital in the design and analysis of optical components used in surveying instruments such as levels and theodolites.

Given Data / Assumptions:

• The lens is thin and centred; principal planes coincide with the optical centre.
• Distances f₁ and f₂ are measured along the principal axis with the usual sign convention.
• Small-angle (paraxial) approximation holds.

Concept / Approach:

The thin-lens equation is 1/f = 1/v + 1/u, where u and v are the algebraic object and image distances. If we denote these two conjugate distances as f₁ and f₂, the identical relation becomes 1/f = 1/f₁ + 1/f₂. This equation captures the reciprocal nature of conjugate distances. Newton’s form (x·x′ = f²) is equivalent when distances are measured from the principal foci instead of the optical centre, but with the current definition (from the optical centre), the reciprocal form is correct.

Step-by-Step Solution:

Verification / Alternative check:

Using Newton’s form with x = u − f and x′ = v − f gives x·x′ = f²; rearrangement with u and v recovers the same reciprocal relation.

Why Other Options Are Wrong:

f = f₁ + f₂ — dimensionally inconsistent for conjugate distances.
f = (f₁ f₂)/(f₁ + f₂) — this is the harmonic mean formula for equivalent focal length of two thin lenses in contact, not for object–image distances.
None of these — incorrect because the standard lens formula applies.

Common Pitfalls:

Sign mistakes (treating one distance as positive instead of negative); mixing Newton’s and Gaussian forms without consistent reference points.

Final Answer:

1/f = 1/f₁ + 1/f₂