As far as I know and have read, the following equation has no answer x^2 + y^2 = -a^2
From the point of view of conic sections, the circle is obtained when the cutting plane is parallel to the plane of the circle producing the cone.
My question is, how is this cut that has no answer?
Because I think there is answer no matter how it is cut.
is it related to complex numbers?
For conic sections, $x$ and $y$ are real numbers. For real $x$ and $y$, $x^2 \geq 0$, $y^2 \geq 0$ and $$ x^2 + y^2 \geq 0 \text{.} $$
But then $-a^2 \leq 0$. The only possibility for a solution is obtained when $a = 0$ and the conic section is a point, a circle with radius $0$.