Derive the formula for spherical mirror

According to the Cartesian system in the above figure.

Object distance (PB) = - u

Image distance (PB’) = - v

Focal length (PF) = - f

Radius of curvature (PC) = - R

From the geometry we can see that in the figure above right triangle ΔABP and ΔA’B’P’ are similar.

∴

∴ ……. (1)

Also ΔABC and ΔA’B’C’ are similar.

∴ ……. (2)

Also

From above figure

CB’ = PC – PB’ = - R - (- v) = - R + v.

And

CB = PB – PC = - u - (- R) = - - - u + R.

From (2)

∴

⇒ R (u + v) = 2uv

Dividing both sides by Ruv we get.

We know that f =

Putting this in the above equation we get

This equation is the ‘Mirror Formula’