Conic sections

Conic sections have a unifying definition: given a point , a line and a real number , the conic having focus , directrix and eccentricity is the locus of all points of the plane such that the ratio between their distance from and their distance from is equal to : .
Change the position of the directrix (by moving points and ), the position of the focus and the eccentricity of the conic to observe what kind of curve you get. Verify that the defining condition on is met for different points . The conic is
  • an ellipse when ,
  • a parabola when , and
  • a hyperbola when .
Observe also that when becomes closer and closer to the ellipse approximates better and better a circle: when the ellipse is defined by means of its two foci, so its eccentricity corresponds to the ratio between the focal distance and the major axis, implies that the two foci are the same point so the ellipse reduces exactly to a circle.