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

582

ANALYTIC GEOMETRY

12. Let P be a point on the minimum ellipse of the hyperboloid (1), ~ 4, and let ( be the common angle which the
two rulings through P make with the z-axis. Prove that
tan P = b1/c, where b1 is the half-length of the diameter of the
minimum ellipse which is conjugate to the diameter through P.
13. Using the result of Ex. 12, show that a hyperboloid of
revolution of one sheet can be generated by the rotation of a
ruling of either set about the axis which does not meet the
surface.
14. Prove that the rulings of one set on a hyperbolic paraboloid intercept proportional segments on two rulings of the
other set.
Loci
15. Find the locus of a point which moves so that its distance from a fixed point bears to its distance from a fixed
plane, not through the point, a constant ratio, k.
Ans. A quadric of revolution which is an ellipsoid, an
elliptic paraboloid, or a hyperboloid of two sheets, according
as k is less than, equal to, or greater than unity.
16. A point moves so that its distance to a fixed point bears
to its distance to a fixed line, not through the point, a constant
ratio. Find its locus.
Exercises 17-19. In connection with these exercises, Exs,
28, 29, p. 522 will be found useful.
17. Find the locus of a point which moves so that its distances to two skew lines are always in the same ratio, k.
18. Prove that a line which is rotated about an axis skew
to it generates a hyperboloid of revolution of one sheet; cf.
Ex. 13.
19. Let L and L' be two fixed skew lines and let M and M'
be two planes, which pass through L and L' respectively and
so move that they are always mutually perpendicular. Find
the locus of their line of intersection.