Director circle
# Director 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 $√(a^2+b^2){\textstyle$ {\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 $√(a^2−b^2){\textstyle$ {\sqrt {a^{2}-b^{2}}}}, 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 $P_i,$ weights $w_i,$ and constant C, one can define a circle as the locus of points X such that

$\sum _{i}w_{i}\,d(X,P_{i})^{2}=C.$

The director circle of an ellipse is a special case of this more general construction with two points $P_1$ and $P_2$ at the foci of the ellipse, weights $w_1$ = $w_2$ = 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 $P_1$ and $P_2$ is a fixed constant r, is another special case, with $w_1$ = 1, $w_2$ = $–r^2,$ and C = 0.

## Related constructions

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

