This applet gives a geometric construction for two conics - hyperbola and ellipse, based on the definitions below. We start with two fixed points and (the vertices of the conics) and two points and (the foci of the conics), on a variable distance from the vertices.
A hyperbola is the locus of all points in the plane such that the absolute value of the difference of the distances from to two fixed points and (the foci) is a constant equal to the distance between the vertices:
An ellipse is the locus of all points in the plane such that the sum of the distances to two fixed points and (the two foci) is a constant, equal to the distance between the vertices:
In this construction the locus is based on a point moving along a circle with a center one of the foci and radius equal to the distance between the vertices.

Press the play button to animate point .

Drag the slider to change the distance from the foci to the vertices.

Consider the two cases -- the foci outside of the vertices and the foci between the vertices.

You can follow the construction steps with the controls at the bottom of the applet.