Definition of a Conic Section and Eccentricity

Rather than always thinking about slicing a 3-D cone, it is far more useful for calculation to have a purely 2-D, plane-geometry definition of a conic — one written entirely in terms of distances, so it can be turned into an algebraic equation.

Definition. A conic section (or simply a conic) is the locus of a point $P$ in a plane such that the ratio of

• the distance of $P$ from a fixed point, to
• the distance of $P$ from a fixed line,

is always the same constant number.

• The fixed point is called the focus of the conic, and is usually denoted $S$.
• The fixed line is called the directrix of the conic, and is usually denoted $d$.

If $M$ is the foot of the perpendicular dropped from $P$ onto the directrix $d$ (so $PM$ is the shortest distance from $P$ to the line), then by definition:

This constant ratio is called the eccentricity of the conic, written $e$. So the defining relationship — the focus–directrix property — is usually written either as

The value of $e$ alone tells you which conic you have, without needing to know anything else about its size or position:

• If $e=1$: the conic is a parabola.
• If $0<e<1$: the conic is an ellipse.
• If $e>1$: the conic is a hyperbola. …