focaldirectrix

Focal-directrix, also called the focus–directrix pair, refers to a fixed point (the focus) F and a fixed line (the directrix) l in the plane. This pair is central to the analytic definition of conic sections. For a given eccentricity e ≥ 0, the locus of points P such that the distance PF from the focus to P is e times the distance from P to the directrix, d(P, l), is a conic with eccentricity e. The defining relation is PF = e · d(P, l).

The value of e determines the type of conic: e < 1 yields an ellipse, e = 1 yields a parabola, and e > 1 yields a hyperbola. When e = 1, the locus consists of all points equidistant from the focus and the directrix, which is the standard parabola. For e ≠ 1, the resulting curve is a nondegenerate conic whose orientation depends on the relative position of F and l.

Coordinate form: If the focus is at (f, 0) and the directrix is the line x = d, the locus satisfies sqrt((x − f)^2 + y^2) = e |x − d|. Squaring gives (x − f)^2 + y^2 = e^2 (x − d)^2, a quadratic equation in x and y that represents the conic.

Origins and use: The focus–directrix description has historical roots in the work of Apollonius of Perga and is a standard alternative to the cone-section definition of conic sections. In applications, the focal-directrix property explains key optical and geometric features of conics, such as their reflective behavior and focal relationships.