Derive Mirror Formula and Magnification

Derivation of the Mirror Formula (1/f = 1/v + 1/u)

Consider a ray diagram for a concave mirror forming a real image. An object AB is placed beyond C. The image formed is A'B'.

Step 1: Consider triangles ΔA'B'P and ΔABP. 
These are similar triangles (by angle-angle similarity, as ∠APB = ∠A'PB'). 
Therefore, A'B' / AB = PB' / PB ... (Equation 1)

Step 2: Consider triangles ΔA'B'F and ΔMPF. (We assume the mirror is thin, so point M is very close to P, and MP is a straight vertical line equal to AB). 
These are also similar triangles. 
Therefore, A'B' / MP = FB' / PF
Since MP = AB, we get A'B' / AB = (PB' - PF) / PF ... (Equation 2)

Step 3: From (1) and (2), we can equate the ratios: 
PB' / PB = (PB' - PF) / PF

Step 4: Apply Sign Convention: 
Object distance PB = -u
Image distance PB' = -v
Focal length PF = -f

Step 5: Substitute into the equation: 
(-v) / (-u) = (-v - (-f)) / (-f)
v / u = (-v + f) / (-f)
-vf = u(-v + f)
-vf = -uv + uf

Step 6: Divide the entire equation by uvf: 
-vf/uvf = -uv/uvf + uf/uvf
-1/u = -1/f + 1/v
Rearranging gives the Mirror Formula: 
1/f = 1/v + 1/u

Derivation of Magnification (m)

Magnification (m) is the ratio of image height (h') to object height (h). 
m = h' / h

From Step 1 above, we already have the relation from triangles ΔA'B'P and ΔABP: 
A'B' / AB = PB' / PB

Apply Sign Convention: 
Object height AB = h (positive) 
Image height A'B' = -h' (negative, as it's inverted) 
Object distance PB = -u
Image distance PB' = -v

Substitute these values: 
(-h') / h = (-v) / (-u)
-h' / h = v / u
h' / h = -v / u

Since m = h' / h, we get the Magnification Formula: 
m = -v / u