What does the equation x^2+y^2=0 represent?
Is it a circle of radius 0? I don't think it is because that really doesn't make sense. Wouldn't it just be the degenerate conic of a point? (i.e. the point (0,0))
Yes it's just the point (0, 0).
There are a few ways you can think about it. I would probably do this...
$\displaystyle \displaystyle \begin{align*} x^2 + y^2 &= 0 \\ y^2 &= -x^2 \end{align*}$
The LHS is nonnegative, the RHS is nonpositive. So that means the only way for the equation to be satisfied is for both sides to be equal to 0.