4.3 Focus–directrix definitions of the non-degenerate conics
Earlier, we defined the conic sections as the curves of intersection of planes with a double cone. One of these conic sections, the circle, can be defined as the set of points a fixed distance from a fixed point.
Here we define the other non-degenerate conics, the parabola, ellipse and hyperbola, as sets of points that satisfy a somewhat similar condition.
These three non-degenerate conics (the parabola, ellipse and hyperbola) can be defined as the set of points P in the plane that satisfy the following condition: the distance of P from a fixed point is a constant multiple e of the distance of P from a fixed line. The fixed point is called the focus of the conic, the fixed line is called its directrix, and the constant multiple e is called its eccentricity.
The different conics arise according to the value e of the eccentricity, as follows.
A non-degenerate conic is
an ellipse if 0 ≤ e < 1,
a parabola if e = 1,
a hyperbola if e > 1.
Note: When e = 0, the ellipse is a circle; the focus is the centre of the circle, and the directrix is ‘at infinity’.