What does the intersection between given surface and a surface that contains the y-axis look like?

Question

So I was trying to do this exercise, but when I thought I figured it out it seems to be that my answer is wrong, could anyone tell me why?

question: Given the surface $S$: $z=x^2-1$ and a surface that contains the $y$-axis. What does the intersection look like with the plane $S$?

• hyperbola
• parabola
• ellipse
• straight line(s)

My reasoning So I started out by drawing $S$ in a $xz$-plane which gave me a parabola. So then I reasoned that if I took a plane e.g. y = 5 then the intersection would still give me a parabola, but somehow the answer is straight line(s). Why is this so, becaus when I put it in GeoGebra it also tells me that it should be a parabola. Is there something wrong with my understanding of the question? PLease let met know.

Thanks in advance