# 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 pointsPof 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 ofPfrom the focusF_{1}to the distance from the linedis 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.