Area of a circle "ds" over the whole circle formula_33 is just the arc length, which is its circumference, so this shows that the area "A" enclosed by the circle is equal to formula_34 times the circumference of the circle. Note that the area of a semicircle of radius r can be computed by the integral formula_35. By trigonometric substitution, we substitute formula_36, hence formula_37 The last step follows since the trigonometric identity formula_42 implies that formula_43 and formula_44 have equal integrals over the interval formula_45, using integration by substitution. But on the other hand, since formula_46, the sum of the two integrals is
Director circle ellipse. The Apollonius circle, the locus of points "X" such that the ratio of distances of "X" to two foci "P" and "P" is a fixed constant "r", is another special case, with "w" = 1, "w" = −"r", and "C" = 0. In the case of a parabola the director circle degenerates to a straight line, the directrix of the parabola. 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