Focus-Directrix Property of an Ellipse with Dandelin Spheres

This Demonstration shows a plane cutting a cone in an ellipse; the two spheres tangent to the plane and the cone meet the cone in two circles. The spheres meet the plane at the foci of the ellipse. The directrices are the lines where the cutting plane meets the planes of the circles of contact.

As a point traces the ellipse, the line segments joining the point to the circles are equal in length to the line segments joining the point to the respective foci. The moving point is also joined to the directrices. Next, to see that the ratio of the distances to the foci and the directrices is constant, the lines are projected vertically up and down (as applicable) to the planes of the circles of contact to form triangles. Finally, since the triangles do not change shape as they move, the ratios must be constant, and the two definitions (cone cut by a plane and constant distance to focus-distance to directrix ratio) are seen to be equivalent. Moreover, the third definition (constant sum of distances to the foci) can also be seen.