Obtain the relation between radius of curvature and focal length of a concave mirror with necessary ray diagram.

Explanation

Consider a ray parallel to principal axis of a concave mirror of small aperture incident at M. The ray reflected along MF as shown. Let `theta` is the angle of incidences. As the aperture is small, angle `theta` is small and arc PM is considered as straight segment.
From the `Delta^(le) MCP`

`tan theta=(PM)/(PC)`
`tan theta=(PM)/(R)`
since `theta` is small,
`tan theta= theta`
`:. theta=(PM)/(R) rarr(1)`
From the `Delta^(le) PFM`
`tan 2theta =(PM)/(PF)`
`tan 2theta = (PM)/(F)`
since `theta` is small, `tan 2theta= 2theta`
`:. 2theta= (PM)/(F)`
`theta=(PM)/(2F)rarr(2)` R=3f
`implies f = R/2`