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By the directrix property

The two lines at distance d = a 2 c {\textstyle d={\frac {a^{2}}{c}}} from the center and parallel to the minor axis are called directrices of the hyperbola (see diagram).

For an arbitrary point P {\displaystyle P} of the hyperbola the quotient of the distance to one focus and to the corresponding directrix (see diagram) is equal to the eccentricity: | P F 1 | | P l 1 | = | P F 2 | | P l 2 | = e = c a. {\displaystyle {\frac {|PF_{1}|}{|Pl_{1}|}}={\frac {|PF_{2}|}{|Pl_{2}|}}=e={\frac {c}{a}}\,.} The proof for the pair F 1, l 1 {\displaystyle F_{1},l_{1}} follows from the fact that | P F 1 | 2 = (x − c) 2 + y 2, | P l 1 | 2 = (x − a 2 c) 2 {\displaystyle |PF_{1}|^{2}=(x-c)^{2}+y^{2},\ |Pl_{1}|^{2}=\left(x-{\tfrac {a^{2}}{c}}\right)^{2}} and y 2 = b 2 a 2 x 2 − b 2 {\displaystyle y^{2}={\tfrac {b^{2}}{a^{2}}}x^{2}-b^{2}} satisfy the equation | P F 1 | 2 − c 2 a 2 | P l 1 | 2 = 0. {\displaystyle |PF_{1}|^{2}-{\frac {c^{2}}{a^{2}}}|Pl_{1}|^{2}=0\.} The second case is proven analogously.

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

(The choice e = 1 {\displaystyle e=1} yields a parabola and if e < 1 {\displaystyle e<1} an ellipse.)