Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

THE HYPERBOLA

145

the frustum * cut from the cone by the planes of C and C, is
of the same length, 2 a, for all points P.
Join P with F. Then PF and PR, being tangents from P
to the same sphere, are equal. Similarly, PF' and PR' are
equal. Hence
FP + F'P = RP + R'P= RR',
or FP + FIP = 2a.
But this locus is by definition an ellipse with its foci
at F and F', and hence the proposition is proved for the case
that M cuts only one nappe, the intersection being a closed
curve.
If the plane M cuts both nappes, but does not pass through
0, it is a little harder to draw the figure, one sphere being
inscribed in the one nappe, the other, in the other nappe.
A similar study shows that here the difference between
FP and F' P is equal to RR', and hence the locus is a
hyperbola.
The parabola corresponds to the case that M meets only one
nappe, but does not cut it in a closed curve. This case is
realized when M does not pass through 0 and is parallel to a
generator of the cone.
Let L be a line which is perpendicular to the axis of the
cone in a point of the axis distinct from the vertex. As a
plane, M, rotates about L, it will cut from the cone all three
kinds of conies. This will still be true if we take, as L, any
line of space which does not pass through the vertex and is
not parallel to a generator.
11. Confocal Conies. Two conies are said to be confocal if
they have the same foci; in the case of two parabolas, we demand, further, that they have the same axis.
* No technical knowledge of Solid Geometry beyond the definitions of
the terms used (which can be found in any dictionary) is here needed.
On visualizing the figure, the truth of the statements regarding the space
relations becomes evident.