Equivalence of Definitions of Hyperbola

Theorem
The following definitions of an hyperbola are equivalent:

Proof
Let $K$ be an hyperbola aligned in a cartesian coordinate plane in reduced form.

Thus its foci are at $\left({\mathop \pm c, 0}\right)$.

Let:


 * the transverse axis of $K$ have length $2 a$
 * the conjugate axis of $K$ have length $2 b$.

From Equation of Hyperbola in Reduced Form, the equation of $K$ is:
 * $\dfrac {x^2} {a^2} - \dfrac {y^2} {b^2} = 1$

This has been derived from the equidistance property of $K$.

From Focus of Hyperbola from Transverse and Conjugate Axis:
 * $c^2 - a^2 = b^2$

and so $K$ can be expressed as:
 * $(1): \quad \dfrac {x^2} {a^2} - \dfrac {y^2} {c^2 - a^2} = 1$

It remains to be shown that $K$ possesses the focus-directrix property.


 * HyperbolaFocusDirectrixEquidistance.png

Let $D$ be the straight line whose equation is $x = \dfrac {a^2} c$.

It will be shown that the locus of points $P = \left({x, y}\right)$ such that:
 * the distance from $P$ to $F_1$ is $\dfrac c a$ of the distance from $P$ to $D$

is precisely equation $(1)$.

We have that:
 * $P F_2 = \dfrac c a d$

where:
 * $d = x - \dfrac {a^2} c$

Thus: