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

534

ANALYTIC GEOMETRY

central conic. Let Mintersect the plane K of D in the line L.*
As coordinate axes in K take Ox parallel to L and Oy perpendicular to L, as shown. The
equation of D, referred to these
axes, is of the form (Ch. XII,
p ~3):
X. o (3) Ax2 + Bxy + Cy2 + F'= O.
I/ ^^ ' Draw in M the rectangular
-<,, \ \ I _ / axes, O'x', O'y',whose projections
kid/ [a Halo K? on K are respectively Ox, Oy.
~L// D- pD Let P: (x, y') be an arbitrary
^ ---— x point of M and let P: (x, y) be
FIG. 2 its projection on K. Then
(4) = x', y= y' cos,
where 0 is the acute angle between M and IK.
Since D is the curve in K into which the section, S, of the
cylinder by Mprojects, the equation of S is obtained from
equation (3) of D by substituting for x and y in (3) their values
as given by (4). Thus the equation of S is
(5) Ax'2 + Bx'y' cos 0 + Cy!2 cos2 + F'- 0.
This equation represents a conic and, furthermore, a conic of
the same type as D, since the discriminant of the quadratic
terms:
B2 cos2 0 -4 AC cos2 0 = (B2 - 4 A C) cos2 0
is of the same sign as the discriminant, B2 - 4 AC, of the
quadratic terms in (3), q. e. d.
It is clear that (5) is independent of the height 00' at which
M cuts the axis of the cylinder. In other words, the sections by
two parallel planes are congruent conics.
Suppose, now, that B2 - 4 AC > 0 and that to F' is given in
turn the values 1, - 1, 0. Then (3) represents in turn a hyper* If M is parallel to K, the section by Mf is congruent to the directrix;
in this case, then, no further proof is required.