Editing Hyperbola (section)

==In Cartesian coordinates==

===Equation===
If Cartesian coordinates are introduced such that the origin is the center of the hyperbola and the ''x''-axis is the major axis, then the hyperbola is called ''east-west-opening'' and
:the ''foci'' are the points <math>F_1=(c,0),\ F_2=(-c,0)</math>,{{sfn|Protter|Morrey|1970|p=310}}
:the ''vertices'' are <math>V_1=(a, 0),\ V_2=(-a,0)</math>.{{sfn|Protter|Morrey|1970|p=310}}
For an arbitrary point <math>(x,y)</math> the distance to the focus <math>(c,0)</math> is <math display="inline">\sqrt{(x-c)^2 + y^2}</math> and to the second focus <math display="inline">\sqrt{(x+c)^2 + y^2}</math>. Hence the point <math>(x,y)</math> is on the hyperbola if the following condition is fulfilled
<math display="block">\sqrt{(x-c)^2 + y^2} - \sqrt{(x+c)^2 + y^2} = \pm 2a \ .</math>
Remove the square roots by suitable squarings and use the relation <math>b^2 = c^2-a^2</math> to obtain the equation of the hyperbola:

<math display="block">\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \ .</math>

This equation is called the [[canonical form]] of a hyperbola, because any hyperbola, regardless of its orientation relative to the Cartesian axes and regardless of the location of its center, can be transformed to this form by a change of variables, giving a hyperbola that is [[congruence (geometry)|congruent]] to the original (see [[#Quadratic equation|below]]).

The axes of [[symmetry (geometry)|symmetry]] or ''principal axes'' are the ''transverse axis'' (containing the segment of length 2''a'' with endpoints at the vertices) and the ''conjugate axis'' (containing the segment of length 2''b'' perpendicular to the transverse axis and with midpoint at the hyperbola's center).{{sfn|Protter|Morrey|1970|p=310}} As opposed to an ellipse, a hyperbola has only two vertices: <math>(a,0),\; (-a,0)</math>. The two points <math>(0,b),\; (0,-b)</math> on the conjugate axes are ''not'' on the hyperbola.

It follows from the equation that the hyperbola is ''symmetric'' with respect to both of the coordinate axes and hence symmetric with respect to the origin.

====Eccentricity====
For a hyperbola in the above canonical form, the [[eccentricity (mathematics)|eccentricity]] is given by

<math display="block">e=\sqrt{1+\frac{b^2}{a^2}}.</math>

Two hyperbolas are [[similarity (geometry)|geometrically similar]] to each other – meaning that they have the same shape, so that one can be transformed into the other by [[translation (geometry)|rigid left and right movements]], [[rotation (mathematics)|rotation]], [[reflection (mathematics)|taking a mirror image]], and scaling (magnification) – if and only if they have the same eccentricity.

===Asymptotes===
[[File:Hyperbel-param-e.svg|250px|thumb|Hyperbola: semi-axes ''a'',''b'', linear eccentricity ''c'', semi latus rectum ''p'']]
[[File:Hyperbola-3prop.svg|300px|thumb|Hyperbola: 3 properties]]
Solving the equation (above) of the hyperbola for <math>y</math> yields
<math display="block">y=\pm\frac{b}{a} \sqrt{x^2-a^2}.</math>
It follows from this that the hyperbola approaches the two lines
<math display="block">y=\pm \frac{b}{a}x </math>
for large values of <math>|x|</math>. These two lines intersect at the center (origin) and are called ''asymptotes'' of the hyperbola <math>\tfrac{x^2}{a^2}-\tfrac{y^2}{b^2}= 1 \ .</math>{{sfn|Protter|Morrey|1970|pp=APP-29–APP-30}}

With the help of the second figure one can see that
:<math>{\color{blue}{(1)}}</math> The ''perpendicular distance from a focus to either asymptote'' is <math>b</math> (the semi-minor axis).

From the [[Hesse normal form]] <math>\tfrac{bx\pm ay}{\sqrt{a^2+b^2}}=0 </math> of the asymptotes and the equation of the hyperbola one gets:<ref name=Mitchell>Mitchell, Douglas W., "A property of hyperbolas and their asymptotes", ''Mathematical Gazette'' 96, July 2012, 299–301.</ref>
:<math>{\color{magenta}{(2)}}</math> The ''product of the distances from a point on the hyperbola to both the asymptotes'' is the constant <math>\tfrac{a^2b^2}{a^2+b^2}\ , </math> which can also be written in terms of the eccentricity ''e'' as <math>\left( \tfrac{b}{e}\right) ^2.</math>

From the equation <math>y=\pm\frac{b}{a}\sqrt{x^2-a^2}</math> of the hyperbola (above) one can derive:
:<math>{\color{green}{(3)}}</math> The ''product of the slopes of lines from a point P to the two vertices'' is the constant <math>b^2/a^2\ .</math>

In addition, from (2) above it can be shown that<ref name=Mitchell/>
:<math>{\color{red}{(4)}}</math> ''The product of the distances from a point on the hyperbola to the asymptotes along lines parallel to the asymptotes'' is the constant <math>\tfrac{a^2+b^2}{4}.</math>

===Semi-latus rectum===
The length of the chord through one of the foci, perpendicular to the major axis of the hyperbola, is called the ''latus rectum''. One half of it is the ''semi-latus rectum'' <math>p</math>. A calculation shows
<math display="block">p = \frac{b^2}a.</math>
The semi-latus rectum <math>p</math> may also be viewed as the ''[[radius of curvature]] '' at the vertices.

===Tangent===
The simplest way to determine the equation of the tangent at a point <math>(x_0,y_0)</math> is to [[implicit differentiation|implicitly differentiate]] the equation <math>\tfrac{x^2}{a^2}-\tfrac{y^2}{b^2}= 1</math> of the hyperbola. Denoting ''dy/dx'' as ''y′'', this produces
<math display="block">\frac{2x}{a^2}-\frac{2yy'}{b^2}= 0 \ \Rightarrow \ y'=\frac{x}{y}\frac{b^2}{a^2}\ \Rightarrow \ y=\frac{x_0}{y_0}\frac{b^2}{a^2}(x-x_0) +y_0.</math>
With respect to <math>\tfrac{x_0^2}{a^2}-\tfrac{y_0^2}{b^2}= 1</math>, the equation of the tangent at point <math>(x_0,y_0)</math> is
<math display="block">\frac{x_0}{a^2}x-\frac{y_0}{b^2}y = 1.</math>

A particular tangent line distinguishes the hyperbola from the other conic sections.<ref>J. W. Downs, ''Practical Conic Sections'', Dover Publ., 2003 (orig. 1993): p. 26.</ref> Let ''f'' be the distance from the vertex ''V'' (on both the hyperbola and its axis through the two foci) to the nearer focus. Then the distance, along a line perpendicular to that axis, from that focus to a point P on the hyperbola is greater than 2''f''. The tangent to the hyperbola at P intersects that axis at point Q at an angle ∠PQV of greater than 45°.

===Rectangular hyperbola===
In the case <math>a = b</math> the hyperbola is called ''rectangular'' (or ''equilateral''), because its asymptotes intersect at right angles. For this case, the linear eccentricity is <math>c=\sqrt{2}a</math>, the eccentricity <math>e=\sqrt{2}</math> and the semi-latus rectum <math>p=a</math>. The graph of the equation <math>y=1/x</math> is a rectangular hyperbola.

===Parametric representation with hyperbolic sine/cosine===
Using the [[hyperbolic function|hyperbolic sine and cosine functions]] <math>\cosh,\sinh</math>, a parametric representation of the hyperbola <math>\tfrac{x^2}{a^2}-\tfrac{y^2}{b^2}= 1</math> can be obtained, which is similar to the parametric representation of an ellipse:
<math display="block">(\pm a \cosh t, b \sinh t),\, t \in \R \ ,</math>
which satisfies the Cartesian equation because <math>\cosh^2 t -\sinh^2 t =1 .</math>

Further parametric representations are given in the section [[#Parametric equations|Parametric equations]] below.

[[File:Conjugate-unit-hyperbolas.svg|thumb|Here {{nowrap|''a'' {{=}} ''b'' {{=}} 1}} giving the [[unit hyperbola]] in blue and its conjugate hyperbola in green, sharing the same red asymptotes.]]
=== Conjugate hyperbola ===
{{Main|Conjugate hyperbola}}
For the hyperbola <math>\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1</math>, change the sign on the right to obtain the equation of the '''conjugate hyperbola''':
:<math>\frac{x^2}{a^2}-\frac{y^2}{b^2} = -1</math> (which can also be written as <math>\frac{y^2}{b^2}-\frac{x^2}{a^2} = 1</math>).

A hyperbola and its conjugate may have [[conjugate diameters#Of hyperbola|diameters which are conjugate]]. In the theory of [[special relativity]], such diameters may represent axes of time and space, where one hyperbola represents [[event (relativity)|event]]s at a given spatial distance from the [[centre (geometry)#Projective conics|center]], and the other represents events at a corresponding temporal distance from the center.

:<math>xy = c^2</math> and <math>xy = -c^2</math> also specify conjugate hyperbolas.