Question

What is the difference between focus and principal focus.

Answer 1

Principal focus is a point (on the principal axis) where parallel incident rays meet after reflecting (in mirrors) or refracting (in lenses). It is a fixed point of particular lens/mirror and depends upon its shape. Principal Focus also determines the focal length of the lens/mirror.

we can say, $f=\frac{R}{2}\left(forparaxialrays\right)$ where R is radius of curvature.

Focus is just a random point where some rays meet, it might be at the principal focus or otherwise. It also need no be placed on the principal axis of the lens/mirror.