Editing Hyperbola (section)

==Properties==

===Reflection property===
[[File:Hyperbel-wh-s.svg|300px|thumb|Hyperbola: the tangent bisects the lines through the foci]]
The tangent at a point <math>P</math> bisects the angle between the lines <math>\overline{PF_1}, \overline{PF_2}.</math> This is called the ''optical property'' or ''reflection property'' of a hyperbola.<ref> {{citation |last1=Coffman |first1=R. T. |last2=Ogilvy |first2=C. S. |year=1963 |title=The 'Reflection Property' of the Conics |journal=Mathematics Magazine |volume=36 |number=1 |pages=11–12 |jstor=2688124 |doi=10.1080/0025570X.1963.11975375 }} {{pb}} {{citation |last=Flanders |first=Harley |year=1968 |title=The Optical Property of the Conics |journal=American Mathematical Monthly |volume=75 |number=4 |page=399 |jstor=2313439 |doi=10.1080/00029890.1968.11970997 }} {{pb}}
{{citation |last=Brozinsky |first=Michael K. |year=1984 |title=Reflection Property of the Ellipse and the Hyperbola |journal=College Mathematics Journal |volume=15 |number=2 |pages=140–42 |jstor=2686519 |doi=10.1080/00494925.1984.11972763 <!-- Deny Citation Bot--> |doi-broken-date=2024-12-16 |url=https://www.tandfonline.com/doi/abs/10.1080/00494925.1984.11972763 |url-access=subscription }} </ref>
;Proof:
Let <math>L</math> be the point on the line <math>\overline{PF_2}</math> with the distance <math>2a</math> to the focus <math>F_2</math> (see diagram, <math>a</math> is the semi major axis of the hyperbola). Line <math>w</math> is the bisector of the angle between the lines <math>\overline{PF_1}, \overline{PF_2}</math>. In order to prove that <math>w</math> is the tangent line at point <math>P</math>, one checks that any point <math>Q</math> on line <math>w</math> which is different from <math>P</math> cannot be on the hyperbola. Hence <math>w</math> has only point <math>P</math> in common with the hyperbola and is, therefore, the tangent at point <math>P</math>. <br/>
From the diagram and the [[triangle inequality]] one recognizes that <math>|QF_2|<|LF_2|+|QL|=2a+|QF_1|</math> holds, which means: <math>|QF_2|-|QF_1|<2a</math>. But if <math>Q</math> is a point of the hyperbola, the difference should be <math>2a</math>.

===Midpoints of parallel chords===
[[File:Hyperbel-psehnen-s.svg|thumb|Hyperbola: the midpoints of parallel chords lie on a line.]]
[[File:Hyperbel-sa-s.svg|thumb|Hyperbola: the midpoint of a chord is the midpoint of the corresponding chord of the asymptotes.]]
The midpoints of parallel chords of a hyperbola lie on a line through the center (see diagram).

The points of any chord may lie on different branches of the hyperbola.

The proof of the property on midpoints is best done for the hyperbola <math>y=1/x</math>. Because any hyperbola is an affine image of the hyperbola <math>y=1/x</math> (see section below) and an affine transformation preserves parallelism and midpoints of line segments, the property is true for all hyperbolas:<br/>
For two points <math>P=\left(x_1,\tfrac {1}{x_1}\right), \ Q=\left(x_2,\tfrac {1}{x_2}\right)</math> of the hyperbola <math>y=1/x</math>

:the midpoint of the chord is <math>M=\left(\tfrac{x_1+x_2}{2},\cdots\right)=\cdots =\tfrac{x_1+x_2}{2}\; \left(1,\tfrac{1}{x_1x_2}\right) \ ;</math>

:the slope of the chord is <math>\frac{\tfrac {1}{x_2}-\tfrac {1}{x_1}}{x_2-x_1}=\cdots =-\tfrac{1}{x_1x_2} \ .</math>

For parallel chords the slope is constant and the midpoints of the parallel chords lie on the line <math>y=\tfrac{1}{x_1x_2} \; x \ .</math>

Consequence: for any pair of points <math>P,Q</math> of a chord there exists a ''skew reflection'' with an axis (set of fixed points) passing through the center of the hyperbola, which exchanges the points <math>P,Q</math> and leaves the hyperbola (as a whole) fixed. A skew reflection is a generalization of an ordinary reflection across a line <math>m</math>, where all point-image pairs are on a line perpendicular to <math>m</math>.

Because a skew reflection leaves the hyperbola fixed, the pair of asymptotes is fixed, too. Hence the midpoint <math>M</math> of a chord <math>P Q</math> divides the related line segment <math>\overline P \, \overline Q</math> between the asymptotes into halves, too. This means that <math>|P\overline P|=|Q\overline Q|</math>. This property can be used for the construction of further points <math>Q</math> of the hyperbola if a point <math>P</math> and the asymptotes are given.

If the chord degenerates into a ''tangent'', then the touching point divides the line segment between the asymptotes in two halves.

===Orthogonal tangents – orthoptic===
[[File:Orthoptic-hyperbola-s.svg|thumb|Hyperbola with its orthoptic (magenta)]]
{{Main|Orthoptic (geometry)}}
For a hyperbola <math display="inline">\frac{x^2}{a^2}-\frac{y^2}{b^2}=1, \, a>b</math> the intersection points of ''orthogonal'' tangents lie on the circle <math>x^2+y^2=a^2-b^2</math>. <br/>
This circle is called the ''orthoptic'' of the given hyperbola.

The tangents may belong to points on different branches of the hyperbola.

In case of <math>a\le b</math> there are no pairs of orthogonal tangents.

===Pole-polar relation for a hyperbola===
[[File:Hyperbel-pol-s.svg|250px|thumb|Hyperbola: pole-polar relation]]
Any hyperbola can be described in a suitable coordinate system by an equation <math>\tfrac{x^2}{a^2}-\tfrac{y^2}{b^2}= 1</math>. The equation of the tangent at a point <math>P_0=(x_0,y_0)</math> of the hyperbola is <math>\tfrac{x_0x}{a^2}-\tfrac{y_0y}{b^2}=1.</math> If one allows point <math>P_0=(x_0,y_0)</math> to be an arbitrary point different from the origin, then

:point <math>P_0=(x_0,y_0)\ne(0,0)</math> is mapped onto the line <math>\frac{x_0x}{a^2}-\frac{y_0y}{b^2}=1 </math>, not through the center of the hyperbola.

This relation between points and lines is a [[bijection]].

The [[inverse function]] maps

:line <math>y=mx+d,\ d\ne 0</math> onto the point <math>\left(-\frac{ma^2}{d},-\frac{b^2}{d}\right)</math> and

:line <math>x=c,\ c\ne 0</math> onto the point <math>\left(\frac{a^2}{c},0\right)\ .</math>

Such a relation between points and lines generated by a conic is called '''pole-polar relation''' or just ''polarity''. The pole is the point, the polar the line. See [[Pole and polar]].

By calculation one checks the following properties of the pole-polar relation of the hyperbola:
* For a point (pole) ''on'' the hyperbola the polar is the tangent at this point (see diagram: <math>P_1,\ p_1</math>).
* For a pole <math>P</math> ''outside'' the hyperbola the intersection points of its polar with the hyperbola are the tangency points of the two tangents passing <math>P</math> (see diagram: <math>P_2,\ p_2,\ P_3,\ p_3</math>).
* For a point ''within'' the hyperbola the polar has no point with the hyperbola in common. (see diagram: <math>P_4,\ p_4</math>).

''Remarks:''
# The intersection point of two polars (for example: <math>p_2,p_3</math>) is the pole of the line through their poles (here: <math>P_2,P_3</math>).
# The foci <math>(c,0),</math> and <math> (-c,0)</math> respectively and the directrices <math>x=\tfrac{a^2}{c}</math> and <math>x=-\tfrac{a^2}{c}</math> respectively belong to pairs of pole and polar.
Pole-polar relations exist for ellipses and parabolas, too.

===Other properties===
* The following are [[concurrent lines|concurrent]]: (1) a circle passing through the hyperbola's foci and centered at the hyperbola's center; (2) either of the lines that are tangent to the hyperbola at the vertices; and (3) either of the asymptotes of the hyperbola.<ref name=web4>{{cite web |title=Hyperbola |website=Mathafou.free.fr |url=http://mathafou.free.fr/themes_en/hyperb.html |access-date=26 August 2018 |url-status=dead |archive-url=https://web.archive.org/web/20160304061843/http://mathafou.free.fr/themes_en/hyperb.html |archive-date=4 March 2016}}</ref><ref name="web1">{{cite web |title=Properties of a Hyperbola |url=http://www.ul.ie/~rynnet/swconics/HP%27s.htm |access-date=2011-06-22 |url-status=dead |archive-url=https://web.archive.org/web/20170202180210/http://www3.ul.ie/~rynnet/swconics/HP's.htm |archive-date=2017-02-02}}</ref>
* The following are also concurrent: (1) the circle that is centered at the hyperbola's center and that passes through the hyperbola's vertices; (2) either directrix; and (3) either of the asymptotes.<ref name=web1/>
* Since both the transverse axis and the conjugate axis are axes of symmetry, the [[symmetry group]] of a hyperbola is the [[Klein four-group]].
* The rectangular hyperbolas ''xy'' = [[constant (mathematics)|constant]] admit [[group action]]s by [[squeeze mapping]]s which have the hyperbolas as [[invariant set]]s.