For paraxial rays, show that the focal length of a spherical mirror is one-half of its radius of curvature.

Large spherical mirror from figure (b) we can be that in care of large spherical mirror the mirror does not have a well defined focal point this is called spherical observed and results in a blurned image of an extended object.From figure (c) we can see that a spherical mirror that is small compared to is radius of curvature the mirror approximation of parabolic mirror.So rays that arrive parallel to the optical axis are reflection to a well defined focal point so we can say for paraxial rays spherical mirror has are defined focus wellHence in figure we can see that a signal rays is reflected by a spherical concave mirror.the incident ray is parallel to optical axis .For paraxial rays all rays will meet at the focus.R=CF+FPand angle of incidence =angle of reflectionsinθ≈θ(small angle approximates)FX≈FPorCF≈FPR=CF+FPFP+FP2FP2×Ff=R2(in the small angles approximation for paraxial approximation ) focal length of a mirror is half times its Radius.