Finding all intersection points of a conic in complex projective space

Question

I was given an exercise that involved finding the intersection of a curve $x^2 + y^2 = z^2$ in $\mathbb{CP}^2$ and the line at infinity ($z=0$). Plugging in $z = 0$ into the equation of the curve I obtained $x^2 = - y^2$. Then setting by $y=1$, I obtained that there are two intersection points $A = (i:1:0)$ and $B = (-i:1:0)$. My question is how to I make sure that these are the only intersection points? Could there be other intersection points if I plug in different values of $x$ and $y$?