Hyperbola as Locus of Points. The hyperbola is all points where the difference of the distances to two fixed points (the focii) is a fixed constant. Here a slider is used to specify the length of a longer segment. Point C between the endpoints of segment B specifies a short segment which stays the same, the fixed constant which is the difference in the definition. Manipulate the slider to see how this works. The two circles have radii determined by the length of the whole long segment and the remaining part of the whole segment after segment a is subracted. Where the circles intersect are the points which satisfy the locus.
Manipulate the points and the slider to see the circles change and trace out the hyperbola.