"Here's What Happened"
at this stage in the course -

Assumptions
• A non-degenerate conic curve (NDCC), C , is the intersection of a plane, $\pi$,  with a cone that does not pass through the vertex of the cone, $O$.
• We can assign 3 dimensional Cartesian coordinates so that $O = (0,0,0)$ and $\pi =\{(x,y,z): z=1\}$.
• C  is a NDCC if and only if  there is a real irreducible polynomial $P(x,y)=Ax^2 + Bxy + Cy^2 + Dx + Ey + F$  with
$A, B$, and $C$  not all 0 AND   C $= \{(x,y,1): P(x,y) = 0\}.$