The ellipse
Let P be an arbitrary point of the conic
section. Consider the two spheres which touch the cone and the
plane. Let sphere G1 and G2 touch
the cone at circles k1 and k2, and
the plane at points F1 and F2,
respectively. Let the generator through P intersect
k1 and k2 at P1
and P2, respectively. As
PP1=PF1 and
PP2=PF2,
PF1+PF2=P1P2.
Since the line segment P1P2 is
bounded by circles k1 and k2, its
length does not depend on the choice of point P. Hence,
For all points P of the conic,
PF1+PF2 is constant; and thus, by
the definition, the conic is an ellipse.
We can use the spheres to prove another important
property of the ellipse. Let the intersection of the planes of circle
k1 and the ellipse be line d, and let
D and P* be the projections of P to
d and to the plane of of circle k1,
respectively. It is easy to see that the triangles
PP*P1 are similar for all choices
of P, thus the ratio of distances PP* and
PP1=PF1 is constant. The triangles
PP*D are also similar, which yields that the
ratio of PP* and PD is another
constant. Putting the results together, we obtain that
The ratio of the distance of P
from the focus F1 to the distance from the line
d is constant.
The constant is less than 1, because angle
P*PP1 is the half of the apex
angle of the cone, and angle P*PD is greater.
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