Editing Hyperbola (section)

===By the directrix property===
[[File:Hyperbel-ll-e.svg|300px|thumb|Hyperbola: directrix property]]
[[File:Hyperbel-ll-def.svg|300px|thumb|Hyperbola: definition with directrix property]]
The two lines at distance <math display="inline">d = \frac{a^2}c</math> from the center and parallel to the minor axis are called '''directrices''' of the hyperbola (see diagram).

For an arbitrary point <math>P</math> of the hyperbola the quotient of the distance to one focus and to the corresponding directrix (see diagram) is equal to the eccentricity:
<math display="block">\frac{|PF_1|}{|Pl_1|} = \frac{|PF_2|}{|Pl_2|} = e= \frac{c}{a} \, .</math>
The proof for the pair <math>F_1, l_1</math> follows from the fact that <math>|PF_1|^2 = (x-c)^2+y^2,\ |Pl_1|^2 = \left(x-\tfrac{a^2}{c}\right)^2</math> and <math>y^2 = \tfrac{b^2}{a^2}x^2-b^2</math> satisfy the equation
<math display="block">|PF_1|^2-\frac{c^2}{a^2}|Pl_1|^2 = 0\ .</math>
The second case is proven analogously.
[[File:Kegelschnitt-schar-ev.svg|thumb|Pencil of conics with a common vertex and common semi latus rectum]]
The ''inverse statement'' is also true and can be used to define a hyperbola (in a manner similar to the definition of a parabola):

For any point <math>F</math> (focus), any line <math>l</math> (directrix) not through <math>F</math> and any [[real number]] <math>e</math> with <math>e > 1</math> the set of points (locus of points), for which the quotient of the distances to the point and to the line is <math>e</math>
<math display="block">H = \left\{P \, \Biggr| \, \frac{|PF|}{|Pl|} = e\right\} </math>
is a hyperbola.

(The choice <math>e = 1</math> yields a [[parabola]] and if <math>e < 1</math> an [[ellipse]].)

====Proof====
Let <math>F=(f,0) ,\ e >0</math> and assume <math>(0,0)</math> is a point on the curve.
The directrix <math>l</math> has equation <math>x=-\tfrac{f}{e}</math>. With <math>P=(x,y)</math>, the relation <math>|PF|^2 = e^2|Pl|^2</math> produces the equations
:<math>(x-f)^2+y^2 = e^2\left(x+\tfrac{f}{e}\right)^2 = (e x+f)^2</math> and <math>x^2(e^2-1)+2xf(1+e)-y^2 = 0.</math>
The substitution <math>p=f(1+e)</math> yields
<math display="block">x^2(e^2-1)+2px-y^2 = 0.</math>
This is the equation of an ''ellipse'' (<math>e<1</math>) or a ''parabola'' (<math>e=1</math>) or a ''hyperbola'' (<math>e>1</math>). All of these non-degenerate conics have, in common, the origin as a vertex (see diagram).

If <math>e > 1</math>, introduce new parameters <math>a,b</math> so that <math>e^2-1 = \tfrac{b^2}{a^2}, \text { and }\ p = \tfrac{b^2}{a}</math>, and then the equation above becomes
<math display="block">\frac{(x+a)^2}{a^2} - \frac{y^2}{b^2} = 1 \, ,</math>
which is the equation of a hyperbola with center <math>(-a,0)</math>, the ''x''-axis as major axis and the major/minor semi axis <math>a,b</math>.

[[File:Hyperbel-leitl-e.svg|thumb|upright=1.4|Hyperbola: construction of a directrix]]
====Construction of a directrix====
Because of <math>c \cdot \tfrac{a^2}{c}=a^2</math> point <math>L_1</math> of directrix <math>l_1</math> (see diagram) and focus <math>F_1</math> are inverse with respect to the [[circle inversion]] at circle <math>x^2+y^2=a^2</math> (in diagram green). Hence point <math>E_1</math> can be constructed using the [[theorem of Thales]] (not shown in the diagram). The directrix <math>l_1</math> is the perpendicular to line <math>\overline{F_1F_2}</math> through point <math>E_1</math>.

''Alternative construction of <math>E_1</math>'': Calculation shows, that point <math>E_1</math> is the intersection of the asymptote with its perpendicular through <math>F_1</math> (see diagram).