Usually in a Cartesian form of derivation in math and physics, I have seen that a particular formula is derived for simplicity by taking concerned points say in the first quadrant. Few examples are the derivation of mirror formula, lens formula, lens maker formula, distance between two points. Later, they extend the formula to all cases and say : "The formula is valid for all cases independent of quadrant and position because we have taken proper care of signs". For example, see the case of mirror formula derivation in the book (especially the pointed last paragraph):
Similarly here for distance formula in SL loney.
I do not understand deeply that why do this happen ? I know that it can be proved in the above cases by actually deriving for all cases and observing that it is confirming, but how can we say it generally? So the question is :
Why do the sign convention in coordinate geometry work the way it does ?