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What are hyperbolas?

What are hyperbolas? Here are some definitions.

Noun
  1. plural of hyperbola
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It's the other conic sections that confuse me, like ellipses and hyperbolas.
In book two Apollonius investigates how hyperbolas are related to their asymptotes, and he also studies how to draw tangents to given conics.
Long-period comets can have orbits ranging from eccentric ellipses to parabolas to even modest hyperbolas.
A family of hyperbolas as shown in the figure may be printed on a chart.
Angeli's many works were on infinitesimals and he used them to study spirals, parabolas and hyperbolas.
The intersections of the surface with planes parallel to the xz and yz planes are hyperbolas.

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