Equivalence of Definitions of Ellipse

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Theorem

The following definitions of an ellipse are equivalent:

Equidistance Property

EllipseEquidistance.png

Let $F_1$ and $F_2$ be two points in the plane.

Let $d$ be a length greater than the distance between $F_1$ and $F_2$.

Let $K$ be the locus of points $P$ which are subject to the condition:

$d_1 + d_2 = d$

where:

$d_1$ is the distance from $P$ to $F_1$
$d_2$ is the distance from $P$ to $F_2$.

Then $K$ is an ellipse.


Focus-Directrix Property

EllipseFocusDirectrix.png


Let $D$ be a straight line.

Let $F$ be a point.

Let $\epsilon \in \R: 0 < \epsilon < 1$.


Let $K$ be the locus of points $b$ such that the distance $p$ from $P$ to $D$ and the distance $q$ from $P$ to $F$ are related by the condition:

$\epsilon \, p = q$


Then $K$ is an ellipse.


Proof

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

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


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 $P = \left({x, y}\right)$ 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:

$P F_2 = \epsilon \left({d - x}\right)$

where:

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

Thus:

\(\displaystyle \sqrt {\left({x - c}\right)^2 + y^2}\) \(=\) \(\displaystyle \frac c a \left({\frac {a^2} c - x}\right)\) $\quad$ $\quad$
\(\displaystyle \implies \ \ \) \(\displaystyle \left({x - c}\right)^2 + y^2\) \(=\) \(\displaystyle \left({a - \frac c a x}\right)^2\) $\quad$ $\quad$
\(\displaystyle \implies \ \ \) \(\displaystyle x^2 - 2 c x + c^2 + y^2\) \(=\) \(\displaystyle a^2 - 2 c x + \frac {c^2} {a^2} x^2\) $\quad$ $\quad$
\(\displaystyle \implies \ \ \) \(\displaystyle x^2 \left({1 - \frac {c^2} {a^2} }\right) + y^2\) \(=\) \(\displaystyle a^2 - c^2\) $\quad$ $\quad$
\(\displaystyle \implies \ \ \) \(\displaystyle \left({a^2 - c^2}\right) \frac {x^2} {a^2} + y^2\) \(=\) \(\displaystyle a^2 - c^2\) $\quad$ $\quad$
\(\displaystyle \implies \ \ \) \(\displaystyle \frac {x^2} {a^2} + \frac {y^2} {a^2 - c^2}\) \(=\) \(\displaystyle 1\) $\quad$ $\quad$

$\blacksquare$