Define focal length of a concave mirror. Prove that the radius of curvature of a concave mirror is double its focal length.

Explanation

Focal length of concave mirror :
The distance between the focus F and the pole P of the mirror is called the focal length of the concave mirror.
Consider a ray AB parallel to principal axis incident on a concave mirror at B and is reflected along BF.The line CB is normal to the mirror.

Let `theta` be the angle of incidence, `angleABC=angleBCP=theta`
Draw `BD_|_CP`
In right angled `Delta^("le")BCD`
`Tan theta=(BD)/(CD)" ..................(1)"`
From `Delta^("le")BFD, " Tan 2"theta=(BD)/(FD)" .........................(2)"`
Dividing eq by eq (1), `(Tan 2theta)/(Tan theta)=(CD)/(FD)" ......................(3)"`
If `theta` is very small, them `tan theta~~ theta` and `tan 2theta~~2 theta` since the aperature of the lens is small
`therefore" The point B lies very close to p."`
`CD~~CP and FD ~~FP`
From eq (3), `(2theta)/(theta)=(CP)/(FP)=(R)/(f) rArr 2 = (R)/(f)`
`R = 2f`