directrices

Directrices are fixed lines associated with conic sections in analytic geometry. They participate in the focal-directrix definition of a conic: a conic is the locus of points P for which the ratio PF / dist(P, d) = e is constant, where F is a fixed focus, d is a fixed line called the directrix, and e is the eccentricity.

For a parabola, the eccentricity e equals 1 and there is a single directrix; for an ellipse and a hyperbola there are two directrices, each paired with a corresponding focus. In standard position, with the major axis on the x-axis and the center at the origin, the directrices of an ellipse or hyperbola are lines perpendicular to the major axis and are given by x = ± a/e, where a is the semi-major axis and e is the eccentricity. The distance from the center to each directrix is a/e, and the foci lie along the major axis at x = ± c with c = ae.

A common example is the parabola y^2 = 4ax, which has focus at (a, 0) and directrix x = -a; for every point on the parabola, the distance to the focus equals the distance to the directrix.

The term directrix comes from Latin, meaning a guide or directing line. In geometry, directrices are central to the definition and many properties of conic sections, and they also appear in optics and other applications where conic figures play a role. See also focus, eccentricity, and conic sections.