In terms of conic sections, how do parabolas and hyperbolas differ? Is a hyperbola just a parabola taken at a 90 degree angle?
The parabola results from intersecting a double cone with a plane parallel to the edge of the cone. The videos show it pretty well early in the video.
If we tilt the plane one way, it will intersect the single nappe of the cone in an ellipse.
Tilt the plane the other way and it will intersect the two nappes in a hyperbola.
Most pictures of this show the hyperbola formed by a plane parallel to the axis of the cone, but that is not necessary.
Here is the MathWorld explanation from Wolfram.
I've tried to show these ideas below.
The black lines shoe the cone viewed straight on.
The blue line is the edge-on view of a plane intersecting the cone in a parabola. Note the parallel to the edge of the cone.
The red line is a plane that makes a hyperbola. Note the triangle that is formed on the side whose interior angle add to less that 2 right angles.
The orange line is a plane that intersects the cone to form an ellipse.