Equivalence of Definitions of Ellipse

Theorem
The following definitions of an ellipse are equivalent:

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

Thus its foci are at $\tuple {\mathop \pm c, 0}$.

Let:
 * the major axis of $K$ have length $2 a$
 * the minor axis of $K$ have length $2 b$.

From Equation of Ellipse 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 Ellipse from Major and Minor Axis:
 * $a^2 - c^2 = b^2$

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

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


 * EllipseFocusDirectrixEquidistance.png

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

It will be shown that the locus of points $P = \tuple {x, y}$ 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:
 * $P F_2 = \epsilon \paren {d - x}$

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

Thus: