a line
A line is produced
if a right circular cone intersects a plane that goes through both nappes of the cone, but not through the vertex, the resulting curve will be a hyperbola
If all three lines are parallel, there are zero points of intersection. If all three lines go through a point, there is one point of intersection. If two lines are parallel and the third one crosses them, there are two. If the three lines make a triangle, there are three points.
Rotation.
If it's parallel to the base, it's a circle. If it doesn't go through the base, it's an ellipse. If it's does, it's hyperbolic/parabolic.
If I understand your description correctly, a line.
A line is produced
An Ellipse
An ellipse is produced.
It will be a hyperbola.
hyperbola
hyperbola
The intersection forms a hyperbola.
CIRCULAR
Then the intersection is a hyperbola.
When it is produced it is processed through a tube, giving it a circular shape.
A traffic signal displaying a steady circular yellow light means that you should use caution when going through the intersection. You have the right away, but watch for other cars.