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

248

ANALYTIC GEOMETRY

Equation (6) is exactly the equation treated in ~ 3. Therefore the theorems of ~ 3 are valid for it and, consequently, for
the original equation (1).
The value of F' as given by (3) can be put in a more convenient form. Multiply the first of the equations (4) by x0,
the second by yo, and add:
2 AxZo + 2 Bxy, + 2 Cyo2 + Dxo + Eyo = 0.
Multiply this equation by - - and add it to (3):
F'= Dxo + 1 Eyo + F.
Finally, substitute the values of x0 and yo as given by (5).
The result is
F'= 4ACF-B2F- AE2- CD + BDE
B2- 4AC
The numerator of the fraction is known as the discriminant
of equation (1) and is denoted by A:
(7) A = 44ACF- B2F - AE2 - CD2 + BDE.
In terms of A, FT has the value
(8) F' = - A -AC
B2 - 4AC
It is clear that if F' = 0, then A = 0, and conversely. In
stating the theorems of ~3 for equation (1) above, we can,
therefore, replace F' 4 0 and F' = 0 by A = 0 and A = 0 respectively. Furthermore, in case B2- 4 AC is negative and F'
and A are not 0, A has the same sign as F'. In this case,
then, AF' (or CF') is positive or negative, according as AA
(or CA) is positive or negative.
We now restate, for equation (1), the theorems of ~ 3.
THEOREM 3. An equation (1) of hyperbolic type:
B2 - 4 ac > 0,
represents a hyperbola, if A, 0. If A = O, it represents two
intersecting straight lines.