When conic sections are taught in high school, the concept of a focus is introduced from the geometric prespective. Well, at least if your teacher is any good. Later, once the algebraic equation of the conic is established, we find that this special geometric point can be in some way derived from the coefficients of the expressions in the algebraic equation for the conic. This is typically derived in school by moving back and forth between arguments involving that of geometric observation and that of algebraic manipulation.
From an algebraic prespective, it seems quite unintuitive to me that there is a special point outside the curve which is so critical to understanding the curves' geometric properties. In a way, I see it as more of a meaningful quantity than a root.
Now, my question is: Using advanced mathematics, what is the algebraic intuition behind the focus? Can we generalize this intuition to talk about foci of higher degree curves? And, are there analogoues to points like focus for higher degree algebraic curves?