Definition:Hyperbola

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Definition

Intersection with Cone

ConicSections-All.png

Let $C$ be a double napped right circular cone whose base is $B$.

Let $\theta$ be half the opening angle of $C$.

That is, let $\theta$ be the angle between the axis of $C$ and a generatrix of $C$.

Let a plane $D$ intersect $C$.

Let $\phi$ be the inclination of $D$ to the axis of $C$.


Let $K$ be the set of points which forms the intersection of $C$ with $D$.

Then $K$ is a conic section, whose nature depends on $\phi$.


ConicSections-Hyperbola.png


Let $\phi < \theta$.

Then $K$ is a hyperbola.

Note that in this case $D$ intersects $C$ in two places: one for each nappe of $C$.


Focus-Directrix Property

HyperbolaFocusDirectrix.png


Let $D$ be a straight line.

Let $F_1$ be a point.

Let $\epsilon \in \R: \epsilon > 1$.


Let $K$ be the locus of points $P$ such that the distance $p$ from $P$ to $D$ and the distance $q$ from $P$ to $F_1$ are related by the condition:

$\epsilon \, p = q$


Then $K$ is a hyperbola.


Equidistance Property

 

HyperbolaEquidistance.png


Let $F_1$ and $F_2$ be two points in the plane.

Let $d$ be a length less than the distance between $F_1$ and $F_2$.

Let $K$ be the locus of points $P$ which are subject to the condition:

$\left\lvert{d_1 - d_2}\right\rvert = d$

where:

$d_1$ is the distance from $P$ to $F_1$
$d_2$ is the distance from $P$ to $F_2$
$\left\lvert{d_1 - d_2}\right\rvert$ denotes the absolute value of $d_1 - d_2$.

Then $K$ is a hyperbola.


Parts of Hyperbola

 

HyperbolaParts.png


Consider a hyperbola $K$ whose foci are $F_1$ and $F_2$.


Major Axis

The major axis of $K$ is the straight line passing through both of the foci of $K$.


Vertex

Each of the points where $K$ intersects the major axis of $K$ is known as a vertex of $K$.


Transverse Axis

The transverse axis of $K$ is the line segment joining the vertices of $K$.


Minor Axis

The minor axis of $K$ is the perpendicular bisector of the transverse axis of $K$.


Center

The center of $K$ is the point where the major axis and minor axis of $K$ cross.

By definition of the major axis and minor axis, this is the point midway between the foci.


Asymptote

The asymptotes of $K$ are the two straight lines which cross at the center of $K$ such that the distance between $K$ and those lines approaches zero as they tend to infinity.


Conjugate Axis

Let $PQ$ and $RS$ be line segments constructed through the vertices of $K$ parallel to the minor axis of $K$ and intersecting the asymptotes of $K$ at $P$, $Q$, $R$ and $S$ as above.

Construct the line segments $PR$ and $QS$.

Let $C_1$ and $C_2$ be the points of intersection of $PR$ and $QS$ with the minor axis of $K$.


The conjugate axis of $K$ is the line segment $C_1 C_2$.


Covertex

Each of the endpoints of the conjugate axis of $K$ is known as a covertex of $K$.


Reduced Form

Let $K$ be a hyperbola embedded in a cartesian coordinate plane.

$K$ is in reduced form if and only if:

$(1)$ its major axis is aligned with the $x$-axis
$(2)$ its minor axis is aligned with the $y$-axis.


HyperbolaReducedForm.png


Also see

  • Results about hyperbolas can be found here.


Linguistic Note

The word hyperbola is pronounced with the stress on the second syllable: hy-per-bo-la.

The plural of hyperbola is properly hyperbolae, but this is considered pedantic, and the usual plural form found is hyperbolas.

The form hyperbolas is used on $\mathsf{Pr} \infty \mathsf{fWiki}$.


Historical Note

Some sources suggest that the word hyperbola was provided by Apollonius of Perga, who did considerable work on establishing its properties.

However, it is also believed that Menaechmus may have used the term, and that it may go back even further than that.


Sources