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

568

ANALYTIC GEOMETRY

values, which, however, always satisfy (5). To obtain the
locus of all the tangent lines L, we have only to eliminate the
auxiliary variables X, /, v, r from equations (2) and (5). Substituting the values of A, /, v as given by (2) into (5) and suppressing the factor 1/r, we get the equation
Zo(x -- x) + Yo(Y - Yo) + o(Z - z0) = 0
a2 b2 c2
which reduces, by virtue of (4), to
(6) a2 -+ - + c2
But this is the equation of a plane. Hence we have the
theorem:
THEOREM 1. The tangent lines at a point Po of an ellipsoid
all lie in a plane.
The plane of the tangent lines at P0 we define as the tangent plane to the ellipsoid at Po. Its equation is given by (6).
EXERCISES
Find for each of the following surfaces the condition that
a line L through a point Po of the surface be tangent to the
surface. Prove the analogue of Theorem 1 and deduce the
equation of the tangent plane at P0.
1. The unparted hyperboloid. 2. The biparted hyperboloid.
3. The elliptic paraboloid. 4. The hyperbolic paraboloid.
5. The cone, Po not being at the vertex.
6. Prove that the tangent plane to a hyperboloid of one
sheet at a point Po is the plane determined by the rulings
which pass through Po. Hence show that the two sets of
rulings found in ~ 4 exhaust all the straight lines on the
surface.
7. The same for a hyperbolic paraboloid.
8. Let Q be a quadric surface, not a cone or a cylinder.
Prove that a plane is tangent to Q if and only if it intersects