Eccentricity of Parabola equals 1

Theorem

The parabola has eccentricity equal to $1$.

Proof

Let $K$ be a conic section.

From the focus-directrix property of a conic section, $K$ is the locus of points $b$ such that the distance $p$ from $b$ to $D$ and the distance $q$ from $b$ to $F$ are related by the condition:

$(1): \quad q = \epsilon p$

where $\epsilon$ denotes the eccentricity.

Now let $K$ be a parabola.

From the focus-directrix property of the parabola, $K$ is the locus of points $P$ such that the distance $p$ from $P$ to $D$ equals the distance $q$ from $P$ to $F$:

$p = q$

This is an instance of $(1)$ where $\epsilon = 1$.

Hence the result.

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): conic (conic section)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): conic (conic section)