Conic sections have a unifying definition: given a point [math]F[/math], a line [math]d[/math] and a real number [math]e>0[/math], the conic having focus [math]F[/math], directrix [math]d[/math] and eccentricity [math]e[/math] is the locus of all points [math]P[/math] of the plane such that the ratio between their distance from [math]F[/math] and their distance from [math]d[/math] is equal to [math]e[/math]: [math]\frac{PF}{dist(P,d)}=e[/math].

Change the position of the directrix [math]d[/math] (by moving points [math]A[/math] and [math]B[/math]), the position of the focus [math]F[/math] and the eccentricity of the conic to observe what kind of curve you get. Verify that the defining condition on [math]P[/math] is met for different points [math]P[/math].[br][br]The conic is[br][list][br][*] an ellipse when [math]0<e<1[/math],[br][*] a parabola when [math]e=1[/math], and[br][*] a hyperbola when [math]e>1[/math].[br][/list][br][br]Observe also that when [math]e[/math] becomes closer and closer to [math]0[/math] the ellipse approximates better and better a circle: when the ellipse is defined by means of its two foci, so its eccentricity [math]e[/math] corresponds to the ratio between the focal distance and the major axis, [math]e=0[/math] implies that the two foci are the same point so the ellipse reduces exactly to a circle.