If you slice two cones vertically, you get a hyperbola:

Here's what geometrically makes a hyperbola a hyperbola:
Just like the ellipse, you start with the two green dots (the foci)... But, there's a big DIFFERENCE with hyperbolas! (Ha! I just made a math pun!)
Check it out:
If you shine a flashlight on a table horizontally, you'll get a hyperbola  well, half anyway:
Also, when a plane (like The Condorde or the Space Shuttle) goes faster than the speed of sound, the result is a sonic boom. The change in air pressure (due to the extreme speed) creates a cone shaped wave that shoots out of the back of the plane. Just like the flashlight, guess what shape the cone hits the ground in?
The sonic boom is heard (and felt) along the red edge of the hyperbola.