Use a geometrical to derive the relation 1/v + 1/u = 1/f for concave mirror or Derive the concave mirror formula

Use a geometrical to derive the relation 1/v + 1/u = 1/f for concave mirror.

Or

Derive the concave mirror formula

Solution

Consider the incidence of ray OX on to a concave mirror from a point object O placed along the principal axis and then suddenly reflected in the direction XI making an angle q with the normal CX.

Ray OP strikes the mirror incident normally at P and thus reflected back along its own path. The point of intersection I of the two reflected rays is the image position.

From ∆ OXC,  α + θ = β …………….(i)

From ∆ CXI,  β + θ = λ ……………… (ii)

Eqn. (i) – Eqn. (ii);  α – β = β – λ

=> α +λ = 2β ……. (iii)

If X is very close to P the α, λ, β are small angles

α ≈ tan α = XP/u;  β ≈ tan β = XP/r and λ ≈  tan λ = XP/v

Substituting  for α,  β, and λ in equation (iii)

XP/u + XP/v = 2XP/r or 1/u + 1/v = 2/r

but r = 2f

Thus, 1/u + 1/v = 1/f

Alternatively