Definition: A real polynomial $f(x,y,z)$ is homogeneous of degree 2 if $f(tx,ty,tz) = t^2 f(x,y,z)$ for all $x,y,z$ and $t \ne 0$.

Suppose $P(x,y)=Ax^2 + Bxy + Cy^2 + Dx + Ey + F$ and
$f(x,y,z)= Ax^2 + Bxy + Cy^2 + Dxz + Eyz + Fz^2$ is the corresponding homogeneous polynomial of degree 2.

Fact: $P(x,y) = f(x,y,1)$ so

$P(x,y) = 0$  if and only if  $ f(x,y,1) = 0$.