So If you have two cones stacked on top of each other like you see in a normal conic section, and the cones are split perfectly in two (1/2 of the diameter of the cone's base), how would you graph the hyperbola? For a normal hyperbola you would have the cones split in a way that the two parts of the hyperbola can't touch each other, but what if they went through the central point?
Thanks!
Converting my comment to an answer, as requested ...
The result is a "degenerate" hyperbola consisting of crossed lines. See Wikipedia's "Degenerate conic" entry.
BTW, the cones need not be "split through the middle"; this degeneracy only requires that the cutting plane pass through the common vertex of the cones.