Mirror formula is the relationship between object distance (u), image distance (v) and focal length.

## Derivation

The figure shows an object AB at a distance u from the pole of a concave mirror. The image A_{1}B_{1} is formed at a distance v 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 A_{1}CB_{1} and D ACB are equal then the third angle

But ED = ABFrom equations (1) and (2)If D is very close to P then EF = PFBut PC = R, PB = u, PB_{1} = v, PF = fBy 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

Equation (5) gives the mirror formula

Let AB represent an object placed at right angles to the principal axis at a distance greater than the focal length f of the convex lens. The image A^{1}B^{1} is formed between O and F_{1} on the same side as the object is kept and the image is erect and virtual.

OF_{1} = Focal length = f

OA = Object distance = u

OA^{1} = Image distance = v

But from the ray diagram we see that OC = AB

From equation (1) and equation (2), we get

Dividing throughout by uvf

Let AB represent an object placed at right angles to the principal axis at a distance greater than the focal length f of the convex lens. The image A^{1}B^{1} is formed beyond 2F_{2} and is real and inverted.

OA = Object distance = u

OA^{1} = Image distance = v

OF_{2} = Focal length = f

OAB and OA^{1}B^{1} are similar

But we know that OC = AB

the above equation can be written as

From equation (1) and (2), we get

Dividing equation (3) throughout by uvf

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