How to derive mirror equation?

Derivation

The figure shows an object AB at a distance āuā from the pole of a concave mirror. The image A_{1}B_{1} is formed at a distance āvā from the mirror. The position of the image is obtained by drawing a ray diagram.

Consider the D A_{1}CB_{1} 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, PB 1 = v, PF = f

By sign convention

PC = -R, PB = -u, PF = -f and PB 1 = -v

Equation (3) can be written as

Dividing equation (4) throughout by uvf we get

Hence, equation (5) gives the mirror formula.