Defining Eccentricity

The eccentricity, denoted $e$, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.

The eccentricity of a conic section is defined to be the distance from any point on the conic section to its focus, divided by the perpendicular distance from that point to the nearest directrix. The value of $e$ is constant for any conic section. This property can be used as a general definition for conic sections. The value of $e$ can be used to determine the type of conic section as well:

• If $e = 1$, the conic is a parabola
• If $e < 1$, it is an ellipse
• If $e > 1$, it is a hyperbola

The eccentricity of a circle is zero. Note that two conic sections are similar (identically shaped) if and only if they have the same eccentricity.

Recall that hyperbolas and non-circular ellipses have two foci and two associated directrices, while parabolas have one focus and one directrix. In the next figure, each type of conic section is graphed with a focus and directrix. The orange lines denote the distance between the focus and points on the conic section, as well as the distance between the same points and the directrix. These are the distances used to find the eccentricity.

Conic sections and their parts

Eccentricity is the ratio between the distance from any point on the conic section to its focus, and the perpendicular distance from that point to the nearest directrix. 

Conceptualizing Eccentricity

From the definition of a parabola, the distance from any point on the parabola to the focus is equal to the distance from that same point to the directrix. Therefore, by definition, the eccentricity of a parabola must be $1$.

For an ellipse, the eccentricity is less than $1$. This means that, in the ratio that defines eccentricity, the numerator is less than the denominator. In other words, the distance between a point on a conic section and its focus is less than the distance between that point and the nearest directrix. 

Conversely, the eccentricity of a hyperbola is greater than $1$. This indicates that the distance between a point on a conic section the nearest directrix is less than the distance between that point and the focus.