Mirror Formula
An expression showing the relation between object distance, image distance and focal length of a mirror is called mirror formula.
Assumptions and Sign conventions
To derive the mirror formula following assumptions and sign conventions are made;
- The aperture of the mirror is small
- Object should be placed on the principal axis in the form of point object
- All distances are measured from the pole of the mirror
- The distances of the real object and real images are taken as positive whereas that of virtual objects and images are taken negative.
- Focal length and radius of curvature of a concave mirror are positive whereas convex mirror negative.
Mirror formula for concave mirror when real image is formed
Let us take a concave mirror of aperture mirror of aperture XY where a light ray AC is travelling parallel to principle axis from object AB to mirror at C and reflect through focus F and pass through A'. Let another light ray pass directly through focus from A to D at mirror and reflect through A'.

From figure,
ΔABF ~ ΔFN'D
AB / DN' = BF/ FN'
AB / A'B' = (BN' - FN')/ FN' [.: DN' = A'B']
Since N' is very close to point p, FN' is similar to FP and BN' is similar to BP
AB / A'B' = BP – FP / FP ………. (i)
Again,
ΔA'B'F ~ ΔFCN
CN / A'B' = FN / B'F
AB / A'B' = FN / B'N – FN [CN = AB]
Since, N is very close to P, B'N is similar to B'P and FN is similar to FP
AB / A'B' = FP / B'P – FP …… (ii)
Equating equation (i) and (ii)
U – f/ f = f/ v-f
(U – f). (v-f) = f2
Uv – uf – vf + f2 = f2
Dividing both sides by uvf we get,
1/f = 1/u + 1/v
In the case of convex mirror

ΔDNF ~ ΔA'B'F
DN / A'B' = NF / B'F
Since N is very closed to C, NF is similar to PF,
= f / f – v
For virtual image,
= - f/ v – f ……. (i)
Similarly, In ΔABC & ΔA'B'N,
AB / A'B' = BC / B'N
= u / -v …. (ii)
Equating equation (i) and (ii),
u / -v = -f / v -f
on solving and dividing by uvf, we get,
1/f = 1/u + 1/v
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