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

QUADRIC SURFACES

561

EXERCISES
1. Find the equations of the rulings which pass through the
point (3, 2, 5) of the hyperboloid of one sheet of Ex. 1, ~ 2.
2. The same for the hyperbolic paraboloid of Ex. 2, ~ 3, the
point on the surface being (9, 2, 4).
Exercises 3-5. Use considerations of symmetry in the proofs.
3. The two rulings through a point P on a principal section
of a hyperboloid of one sheet are equally inclined to the plane
of the section and lie in a plane M which is perpendicular to
the plane of the section.
4. The same for a hyperbolic paraboloid.
5. If P and P' are points of a hyperboloid of one sheet
which are symmetric in the center, the rulings through P are
parallel to those through P'.
6. Assuming Th. 1, ~ 7, prove that the plane M of Ex. 3
passes through the tangent line at P to the principal section.
Hence show that the projections of the rulings of either set
on a principal plane are the tangents to the principal section
in that plane.
7. The same for a hyperbolic paraboloid, applying the results of Ex. 4.
8. Prove that the plane determined by two parallel rulings
of a hyperboloid of one sheet is tangent to the asymptotic cone
along the element which is parallel to the two rulings.
9. Prove that there are no straight lines on (a) an ellipsoid;
(b) a hyperboloid of two sheets; (c) an elliptic paraboloid.
5. Parallel Sections. Equations (1), (3), and (6), ~ (2), of a
hyperboloid of one sheet, H1, of the conjugate hyperboloid of
two sheets, H2, and of the common asymptotic cone, C, can be
written as the one equation
X2 y2 z2
(1) b2 =
a2 b2 C2