[color=#ff00ff][b]Definition of a Hyperbola: [br][/b][/color][br]A[color=#ff00ff][b] hyperbola is a locus (set of points that satisfy a certain condition) in the plane, for which the difference of the distances from 2 other fixed points, called foci, is constant.[/b][/color] In the applet, [i]A[/i] and [i]B[/i] are foci. Hyperbolas have 2 branches. This applet only traces out one of these branches. [br][br]1) Since the radius of a circle is constant, we know that [i]AC[/i] = constant. [br]2) Thus, this implies [i]AD - CD [/i]= a constant value. [br]3) Since p is the perpendicular bisector of the segment with endpoints [i]B[/i] and [i]C[/i], we know [i]BD[/i] = [i]CD. [br] (Why?)[br][/i]4) Thus, given (2) and (3), we can say that [i]AD[/i] - [i]BD[/i] = a constant value.