Hyperbola

Builds on Euclidean plane Euclidean distance

Definition

Let $F_1$ and $F_2$ be distinct points of the Euclidean plane $\mathbb{R}^2$ , and let $d$ be the Euclidean distance. Let $a$ be a real number with $0 < 2a < d(F_1, F_2)$ . The hyperbola with the foci $F_1$ , $F_2$ and the semi-major axis $a$ is the set $\{\, P \in \mathbb{R}^2 \;:\; |d(P, F_1) - d(P, F_2)| = 2a \,\}.$

Remarks

Let $c = \frac{1}{2} d(F_1, F_2)$ and $b = \sqrt{c^2 - a^2}$ . For $F_1 = (-c, 0)$ and $F_2 = (c, 0)$ , the hyperbola is the set of the points with $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ . This hyperbola has the two asymptotes $y = \pm \frac{b}{a} x$ . Its eccentricity is $e = c / a > 1$ . For $a = b$ , the asymptotes are perpendicular, and the hyperbola is rectangular. The graph of $y = 1/x$ is a rectangular hyperbola.

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Euclidean distance concept
Euclidean plane object
Hyperbola definition

Tag	`0P234N3M28`
URI	`m2://0P234N3M28`
Kind	definition
Writing	W1 stated
MSC	51N20
Defines	Hyperbola
Uses	Euclidean plane, Euclidean distance

Hyperbola

Definition

Remarks

Reading path

Further reading

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