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

CERTAIN GENERAL METHODS

167

Example. Find the equation of the line L which goes
through the point of intersection of the lines (1) and (2) and
cuts the axis of y in the point (0, - 4).
The required line, L, is one of the lines (3); i.e. for a suitable value of c, (3) will represent L. To find this value of k,
we demand that (3) contain the given point (0, -4) of L.
We have, then, setting x = 0 and y =-4 in (3):
(O-4 -2)+k(0+4)=0 or =.
Consequently, the equation of the line L is
x +- y-2 + (x - y)= 0 or 5x- y-4 = 0.
That the line represented by the latter equation does actually
go through the points (1, 1) and (0, -4) can be verified
directly.
The principle which has been set forth for two straight lines
evidently applies to any two intersecting curves whatever, so
that we are now in a position to state the following general
theorem.
THEOREM 1. Let u = 0 and v = 0 be the equations of any
two intersecting curves. Then the equation
u + cv = 0, kc = 0,
represents, in general,* a curve which passes through ill the
points of intersection of the two given curves, and has no other
point in common with either of them.
The last statement in the theorem is new. To prove it,
we have but to note that, if the coordinates of a point P satisfy
the equation u + kv = 0 and also, for example, v = 0, they
must satisfy the equation u = 0; that is, if P is a point on the
curve u + kv = 0, which lies on one of the given curves, it
lies also on the other and so is a point of intersection of the
two.
* It may happen in special cases that the locus u + kv = 0 reduces to
a point, as when, for example,
u = 2x2 + 2y2 - x, v= x2 + y2 -x, =-1.