When I was first introduced parabolas/hyperbolas, circles, and ellipses, I was shown how each and every one of them could be represented as conic sections - an intersection of a plane and a conic surface.
It made me wonder, whether the quadratic equations of parabolas, etc. themselves were actual equations of the intersection of said surfaces, or simply called conic sections because they looked the part.
In the case of the former, would someone please be kind enough to explain how it is so in simple enough terms so a person not so versed in the discipline could grasp the basics?