Why is a hyperbola considered a conic section?
Conic sections are the intersections of a plane and a cone.
When you cut the cone with a plane that is parallel to the base of the cone, you end up with a circle.
When you cut the cone with a plane that is not parallel to the base of the cone and the plane does not cut through the base, you end up with an ellipse. If the plane cuts through the base, you end up with a parabola.
In the case of the hyperbola, you need 2 cones with their bases parallel and away from each other. When your plane cuts through both cones, you have a hyperbola.