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

552

ANALYTIC GEOMETRY

whose equations are obtained by eliminating y from (4) and
(1), and from (4) and (3). But these cylinders are conjugate
hyperbolic cylinders whose common asymptotic planes are
(5) a2+ b- c= 0.
\a2 b2 c2
Consequently, by Ch. XXII, ~ 6, Th. 2, their sections by the
plane (4) are conjugate hyperbolas whose common asymptotes,
defined by equations (4) and (5), pass through
the origin, q. e. d.
The equation of the locus of these asym\j\ / ptotes, as the plane (4) rotates about the axis
\\' % of z, is obtained by eliminating m (now an
auxiliary variable expressing the motion of
y the plane) from equations (4) and (5). The
result is
x2 yz _ 2 z
(6) + 0.
(a2 b2 C2
^ FIG. The locus of the asymptotes is, therefore, a
FIG. 5
cone whose vertex is at the origin. This
cone is called the asymptotic cone of each of the hyperboloids
(1) and (3). Evidently (1) lies wholly without the cone, that
is, on the convex side of it, while (3) lies wholly within it.
EXERCISES
Construct the following hyperboloids, drawing accurately
the principal sections which exist and the sections parallel to
one principal plane.
x2 y2 z z 2x2 2 y 2
1. Z+- = 1. 2. +4- - =-1.
9 4 25 9 4 25
3. 92 _ 16y2+ 36z2 = 144. 4. - 4 x2+ 36y2 _ 9z2 = 36.
5. What is the' equation of the hyperboloid conjugate to the
hyperboloid of Ex. 3? of Ex. 4? Give also in each case the
equation of the common asymptotic cone.