Question

why do we say that a ray passing through the centre of curvature always retraces it's path?

Answer 1

A ray of light passing through the centre of curvature of a concave mirror retraces its path (gets reflected along the same path), because as the ray of light passes through centre of curvature of a concave mirror it strikes the mirror along the normal i.e. it incidences on to the mirror at 90 degree. Hence the incident ray coincides with the normal.
 
 Therefore angle of incidence = 0 so, as we know according to law of reflection
angle of reflection = 0, hence the angle of reflection too become zero degree, thus ray of light retraces its path.