Derive the relation `f = R//2` in the case of a concave mirror.

Explanation

When a parallel beam of light is incident on a concave mirror, the rays are paraxial , i.e. they are incident at a point M , and make a small angle with principal axis. The reflected ray converges at a F on the principal axis. The point F is called the focus of the mirror ,the distance between the principal focus and the pole is called focal length (f).
Consider a ray parallel to the principal axis stricking the mirror at M, then
`angleMFD=angleCMF+angleMCF=theta+theta=2theta`
`angleMFD` is the external angle of the `angleMCF`
In `triangle^"le"` MCD , `tan theta ="MD"/"CD"`...(1)
In `triangle^"le"` MFD , tan `2theta ="MD"/"FD"` ...(2)
`theta` is small , So `tan theta~~ theta , tan 2 theta ~~ 2 theta` ,
Substituting tan `theta` and tan `2theta` in the equations,
`theta="MD"/"CD", 2theta="MD"/"FD"`
`2xxcancel(MD)/(CD)=cancel(MD)/(FD) rArr 2/(CD)=1/(FD)` on CD =2FD
Point D is very close to point P.
`therefore DF ~~ PF ~~ f` and DC=PC=R
`2/(CP)=1/(FP)` CP=3FP
R=2f
`f=R/2`