Eccentricity of Conic Sections
The eccentricity of a conic section describes the shape and “flatness” of the curve. It is defined as:
$$ e = \frac{\text{distance to focus}}{\text{distance to directrix}} $$
This value fully determines the nature of the conic:
- Circle: $e = 0$
- Ellipse: $0 < e < 1$
- Parabola: $e = 1$
- Hyperbola: $e > 1$
1. Eccentricity of an Ellipse
For the ellipse $$ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1, $$ the first eccentricity is:
$$ e = \frac{c}{a} \qquad \text{where } c^2 = a^2 - b^2 $$
The eccentricity describes flatness:
- $e = 0$: ellipse becomes a circle
- small $e$: round ellipse
- $e$ close to 1: very flat, elongated ellipse
2. Second Eccentricity of an Ellipse
An alternate measure of shape is the second eccentricity:
$$ e' = \frac{c}{b} $$
Using $c^2 = a^2 - b^2$, we may also write:
$$ e' = \sqrt{\frac{a^2}{b^2} - 1} $$
3. Distance From Center to the Directrix of an Ellipse
The directrices of the ellipse lie at the distance:
$$ d = \frac{a^2}{c} = \frac{a}{e} $$
measured horizontally from the center.
Derivation
Start with the eccentricity definition:
$$ e = \frac{c}{a}. $$
A point $P$ on the ellipse satisfies:
$$ \frac{\text{distance to focus}}{\text{distance to directrix}} = e. $$
Evaluating this at the vertex $(a,0)$ gives:
$$ \frac{a - c}{\frac{a^2}{c} - a} = \frac{c}{a}, $$
which simplifies to confirm the directrix location:
$$ x = \frac{a^2}{c}. $$
Thus:
$$ d = \frac{a^2}{c} = \frac{a}{e}. $$