Editing Hyperbola (section)

===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>