Is the empty set considered a conic?

Question

If the graph of the equation $Ax^2+Cy^2+Dx+Ey+F=0$ is the empty set (no point at all), is it considered a conic? Isn't a conic supposed to be the intersection of a plane and a cone?

Answer 1

The graph is empty only if you're dealing with strictly real coordinates. Otherwise there's always a solution by the fundamental theorem of algebra. So you see, it all depends on context ($\mathrm R$ or $\mathrm C$) whether you consider all such equations as representing conics, or only those having real points.