The way I learned analytic geometry in highschool, I was given a lot of formulas and shown what they do and maybe, just maybe, why they are accurate. So, an ellipse was defined as being a curve that has the formula $\frac{x^2}{a^2} + \frac{y^2}{b^2} = c^2$ rather than a curve in the plane around two focal points such that the sum of the distances from the focal points to the curve is constant. The latter was presented as a property of the former.
I believe that to properly learn mathematics, you have to do what is done in later courses like real analysis and abstract algebra. That is, you start off with some definitions, and then you deduce the rest. So, in my quest to relearn mathematics in a rigorous way, I have to relearn analytic geometry as well.
I thus have to ask for recommendations for books about analytic geometry.
I’ll suggest you a slogan : “Back to basic !“
You have to see Hilbert’s axioms (1899) :
complemented with e.g.
The “ingredient” for a rigorous treatment of analytical geometry are :
the axiomatization of euclidean geometry
its model in term of the field of real numbers $\mathbb R$.