Eccentricity and Directrix

An alternative way to describe a conic section involves the directrices, the foci, and a new property called eccentricity. The value of a conic’s eccentricity can uniquely identify which type of conic section it is.

The directrix of a conic section is the line that, together with the focus, helps define the conic. Parabolas have one focus and one directrix, while ellipses and hyperbolas have two foci and two associated directrices. The three conic sections with their directrices appear in the following figure.

Let’s verify these eccentricity relationships by examining each conic type.

For parabolas, by definition, the distance from any point on a parabola to the focus equals the distance to the directrix. Therefore, [latex]e = \frac{\text{distance to focus}}{\text{distance to directrix}} = 1[/latex].

For ellipses, consider a horizontal ellipse with directrices at [latex]x = \pm \frac{a^2}{c}[/latex]. The right vertex is at [latex]\left(a,0\right)[/latex] and the right focus is [latex]\left(c,0\right)[/latex]. Therefore the distance from the vertex to the focus is [latex]a-c[/latex] and the distance from the vertex to the right directrix is [latex]\frac{{a}^{2}}{c}-c[/latex]. This gives the eccentricity as:

[latex]e = \frac{a-c}{\frac{a^2}{c}-a} = \frac{c(a-c)}{a^2-ac} = \frac{c(a-c)}{a(a-c)} = \frac{c}{a}[/latex]

Since [latex]c < a[/latex] for ellipses, we have [latex]e < 1[/latex].

For hyperbolas, the directrices are also at [latex]x = \pm \frac{a^2}{c}[/latex], and similar calculations give [latex]e = \frac{c}{a}[/latex]. However, for hyperbolas [latex]c > a[/latex], so [latex]e > 1[/latex].