Definition:Conic Section/Focus-Directrix Property/Circle

Definition

It is not possible to define the circle using the focus-directrix property.

This is because as the eccentricity $e$ tends to $0$, the distance $p$ from $P$ to $D$ tends to infinity.

Thus a circle can in a sense be considered to be a degenerate ellipse whose foci are at the same point, that is, the center of the circle.

Focus

As a circle cannot be specified using the focus-directrix property, the focus cannot be defined using that technique.

Consider an ellipse $K$ with eccentricity $e$.

Let $e$ tend to zero.

Then as $e$ becomes smaller, the foci of $K$ become closer together.

In the limit, the foci coincide.

Thus the focus of the circle can be understood as being its center.

Historical Note

The focus-directrix definition of a conic section was first documented by Pappus of Alexandria.

It appears in his Collection.

As he was scrupulous in documenting his sources, and he gives none for this construction, it can be supposed that it originated with him.

Sources

• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.8$: Pappus (fourth century A.D.): Appendix: The Focus-Directrix-Eccentricity Definitions of the Conic Sections: Footnote
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): circle
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): circle