Derive. The mirror formula

The object distance (u), image distance (v) and focal length (f).

The following sign convention is used for measuring various distances in the ray diagrams of spherical mirrors:

• All distances are measured from the pole of the mirror.

• Distances measured in the direction of the incident ray are positive and the distances measured in the direction opposite to that of the incident rays are negative.

• Distances measured above the principal axis are positive and that measured below the principal axis are negative.
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The figure shows an object AB at a distance u from the pole of a concave mirror. The image A₁B₁ is formed at a distance v from the mirror. The position of the image is obtained by drawing a ray diagram.

Consider the D A₁CB₁ and D ACB

[when two angles of D A₁CB₁ and D ACB are equal then the third angle 

But ED = AB

From equations (1) and (2)

If D is very close to P then EF = PF

But PC = R, PB = u, PB₁ = v, PF = f

By sign convention

PC = -R, PB = -u, PF = -f and PB₁ = -v

 Equation (3) can be written as

Dividing equation (4) throughout by uvf we get