Definitions:
An ellipse is the set of points T such that the sum of the segments F1T and F2T is constant.
A hyperbola is the set of points T such that the absolute value of the difference between F1T and F2T is constant.
To do and explore:
Doubble click the immage and use the tools to animate the construction steps. Construction starts with two points F1 and F2 and a directrix circle having its center in F1. A directrix point P is located on the circle, and we draw the segment F2P. Then we construct the perpendicular bisector L1 and draw the line L2 through F1 and P. Locus of the intersection point T will form an ellips, which is easily shown.
You can move P around the directrix circle. Note that L1 is a tangent line of the ellipse and that T is the tangent point.
Next we draw the segment F2T. By use of congruent triangles and equal angles we can see that a ray from F2 reflected in the ellipse will pass through F1.
Finally we draw a point T' and the segments F1T' and F2T'. We measure these segments and add their lengths. If you move T' along the ellipse, you will see that this sum is constant. Sinse F2T=TP, this sum equals the radius of the directrix circle.
The next experiment works better in the true Cabri program than in CabriJava. The Cabri file is named Directrix_circle.fig.
If you drag the point F2 outside the directrix circle, the locus of T will form a hyperbola. We can for example see that TF2=TP, which can be used to prove that the difference between F1T and F2T is constant. We can also see that a ray from F2 reflected in the hyperbola will be directed from F1.