Editing Hyperbola (section)

===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>.