Definition of the hyperbola

The hyperbola is the third non-degenerate conic. Its definition mirrors that of the ellipse, but with one decisive difference: instead of the sum of the distances from the two foci one considers their difference (in absolute value).

Definition — Hyperbola

A hyperbola is the locus of points $P$ in the plane such that $\bigl| d(P,F_1) - d(P,F_2) \bigr| = 2a, \qquad 2a < d(F_1, F_2),$ with $F_1, F_2$ two fixed points called foci.

The condition $2a < d(F_1, F_2)$ is the opposite of the one for the ellipse and guarantees that the locus exists. Indeed, by the triangle inequality one always has $\bigl| d(P,F_1) - d(P,F_2) \bigr| \le d(F_1, F_2),$ with equality only for the points on the line through $F_1$ and $F_2$: if we required a difference greater than or equal to $d(F_1,F_2)$ we would find no point at all.

Links

Topics: Homographic hyperbola
Concepts: Triangle inequality · Foci · Hyperbola · Locus
Functions: Hyperbola