$\displaystyle x^2+y^2-2x+4y=-6$
?
you could have written: $\displaystyle (x - 1)^2 + (y + 2)^2 = -1$
this is not a circle (unless you made an error), since we could not have a negative number on the right hand side if it were a circle.
unless maybe the circle has a radius in the complex plane, but i don't think that makes sense, never did complex analysis
Here was the original problem statement: CLICK HERE
well, it's none of those if it's of the form $\displaystyle (x - h)^2 + (y - k)^2 = c$ where $\displaystyle c < 0$
so maybe "none" is an option. i vaguely remember something about "degenerate conics" but i'm not sure if this applies here. could you say "degenerate circle" i don't know and very much doubt it