 Vectors and conics

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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.

Eccentricity

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’.

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