The four defining conditions above can be combined into one condition that depends on a fixed point F i.e. the focus, a line L (the directrix) not containing F and a non-negative real number e (the eccentricity). The corresponding conic section consists of all points whose distance to F equals e times their distance to L.
For 0 < e < 1 we obtain an ellipse,
for e = 1 a parabola, and
for e > 1 a hyperbola.
The distance from the center to the directrix is a / e, where a is the semi-major axis of the ellipse, or the distance from the center to the top of the hyperbola. The distance from the center to a focus is ae.
In the case of a circle, the eccentricity e = 0, and the directrix is infinitely far removed from the center.
The eccentricity of a conic section is thus a measure of how far it deviates from being circular.
For a given a, the closer e is to 1, the smaller is the semi-minor axis.
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