Eccentricity of Parabola equals 1
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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)