Establish the relationship between object distance

Establish the relationship between object distance u, image distance v and focal length f of a spherical mirror.

Let us consider an object 00’ placed on the principal axis of a concave mirror (for Convinience of explanation the mirror is drawn as plane mirror but treat the mirror as concave mirror) of focal length ’f.

When, the object lies beyond the ‘C’ ‘a real and inverted image II’ is formed between C and F after reflection from the concave mirror.
From similar triangles BO’A and BFP we can write the proportionalities for corresponding sides as

AB/PB = AO’/PF
====> AB/ AO’ =PB/PF ------(1)

From similar traiangles AI’B and AFP
we can also write,

AB/PS = BI/PF
(or) AB/BI’ = PA/PF-------(2)

From (1) + (2) ,
AB/AO’ +AB/BI’ = PB/PF + PA/PF
=PB+PA/PF = AB/PF

1/AO’ + 1/BI’ = 1/PF
Using cartesian sign convention
AO’ = Object distance = -u
BI’ = Image distance = -v
PF = Focal length = -f
1/-u + 1/-v = -1/f

1/u + 1/v =1/f
This is the mirror formula.