The hyperbola
Let P be an arbitrary point of the
conic. Consider the two spheres which touch the cone and the
plane. Let sphere G_{1} and G_{2} touch
the cone at circles k_{1} and k_{2}, and
the plane at points F_{1} and F_{2},
respectively. Let the generator through P intersect
k_{1} and k_{2} at P_{1}
and P_{2}, respectively. As
PP_{1}=PF_{1} and
PP_{2}=PF_{2},
PF_{1}PF_{2}=P_{1}P_{2}.
The line segment P_{1}P_{2} is bounded
by circles k_{1} and k_{2}; its length
does not depend on the choice of point P. Hence,
For all points P of the conic,
PF_{1}PF_{2} is constant; and thus,
by the definition, the conic is a hyperbola.
We can prove another important property of the
hyperbola. Let the intersection of the planes of circle
k_{1} and the hyperbola be line d, and let
D and P^{*} be the projections of P to
d and to the plane of of circle k_{1},
respectively. It is easy to see that the triangles
PP^{*}P_{1} are similar for all choices
of P, thus the ratio of distances PP^{*} and
PP_{1}=PF_{1} is constant. The triangles
PP^{*}D are also similar, which yields that the
ratio of PP^{*} and PD is another
constant. Putting these results together we obtain that
The ratio of the distance of P
from the focus F_{1} to the distance from the line
d is constant.
The constant is greater than 1, because angle
P^{*}PP_{1} is the half of the apex
angle of the cone, and angle P^{*}PD is smaller.
Back to the previous page
