Question

what is the relationship between sign convention and the mirror equation

Answer 1

Sign Convention are the laws that we should follow to select a positivr or negative sign before a quantity such as focal length, magnification, initial distance, and so on.

Mirror equation is the relationship between focal length, initial distance and final distance.
But, mirror formula of a convpcave mirror is 1/f = 1/v - 1/u

And, for that of covex lens is 1/f = 1/v + 1/u

Here, sign convention is used to differentiate between two kinds of mirrors. But, the question you have asked has caused due to lack of conceptual understanding.