# Director circle

(Redirected from Fermat–Apollonius circle)

In geometry, the director circle of an ellipse or hyperbola (also called the orthoptic circle or Fermat–Apollonius circle) is a circle consisting of all points where two perpendicular tangent lines to the ellipse or hyperbola cross each other.

## Properties

The director circle of an ellipse circumscribes the minimum bounding box of the ellipse. It has the same center as the ellipse, with radius ${\sqrt {a^{2}+b^{2}}}$ , where $a$ and $b$ are the semi-major axis and semi-minor axis of the ellipse. Additionally, it has the property that, when viewed from any point on the circle, the ellipse spans a right angle.

The director circle of a hyperbola has radius a2 - b2, and so, may not exist in the Euclidean plane, but could be a circle with imaginary radius in the complex plane.

## Generalization

More generally, for any collection of points Pi, weights wi, and constant C, one can define a circle as the locus of points X such that

$\sum w_{i}\,d^{2}(X,P_{i})=C.$ The director circle of an ellipse is a special case of this more general construction with two points P1 and P2 at the foci of the ellipse, weights w1 = w2 = 1, and C equal to the square of the major axis of the ellipse. The Apollonius circle, the locus of points X such that the ratio of distances of X to two foci P1 and P2 is a fixed constant r, is another special case, with w1 = 1, w2 = −r2, and C = 0.

## Related constructions

In the case of a parabola the director circle degenerates to a straight line, the directrix of the parabola.