What Is a Hyperbola?

How the branches are organised

As a conic section, the hyperbola is produced when a plane cuts both nappes of a double cone. In analytic geometry, its graph is strongly shaped by its transverse direction and its asymptotes, which guide the branches without ever being touched.

This page keeps the branches, center, and asymptote structure visible together so hyperbola is read as a coherent conic rather than as two disconnected curves.

The branches of a hyperbola do not meet because the defining distance difference stays fixed rather than summing to a constant. The center helps organise the branches, while the asymptotes provide a guide for their direction. Treating those as separate roles makes the graph much easier to interpret.

A hyperbola can open left and right or up and down depending on its transverse direction. The graph's center and asymptotes reveal that orientation before you calculate individual points. The two branches belong to one conic even though a gap remains between them.

• A hyperbola is the set of points whose distances from two foci have a constant difference.
• A hyperbola has two branches and two foci.
• Its asymptotes pass through the center and show the directions the branches approach.
• The difference-of-distances definition separates hyperbola from the sum-of-distances rule for ellipses.

Where hyperbolas help

• Use hyperbolas in coordinate-conic problems involving standard form, center, vertices, foci, and asymptotes.
• Use them when comparing open conics and identifying which one has two branches instead of one.
• Use hyperbola graphs in modelling and analytic settings where asymptotic behavior matters.

Hyperbola checks to make

• Do not confuse the hyperbola's difference-of-distances definition with the ellipse's sum-of-distances definition.
• Do not draw the branches crossing the asymptotes; the graph approaches them but does not meet them in the usual model.
• Do not forget that the center of the hyperbola lies midway between the branches and the foci.