An elementary treatise on cubic and quartic curves, by A. B. Basset.

FOCI OF CARTESIANS.

185

if m and a be eliminated from (21), the result is (23) with c
substituted for p. If however the roots of (23) are complex and
equal to P + iQ, the equations of the two tangents drawn from one
of the circular points are
x+ty=P + Q, +y = P-tQ,
whilst the equations of the two tangents drawn from the other
circular point are
x- y=P+ Q, x-y = P- Q.
These four straight lines intersect in the points
x=P+tQ, y=O,
x = P, y = Q,
x=P, y=Q,
x=P, y= - Q
x=P-iQ, y=O.
Hence there are two imaginary foci which lie on the axis of x,
and two real ones which are determined by the equations
x=P, y=~Q.
The latter foci together with the origin are the only real single
foci which the curve possesses.
277. The coordinates of the points where (22) cuts the axis
of x are determined by the equation
{x2 + 2(f/+ A) x+82} {x2+2(f-A)x +2 = 0,
and the condition that the values of x, obtained by equating both
factors to zero, should be real is that
(8 +f+ A)(8 -f- A) and (8 +f- A) (8 -f+ A)...(24)
should be both negative. Now the condition that the roots of
(23) should be real is that
( +f + A) ( +f — A) ( -f + A) (6 -f- A)
should be positive. Hence it follows from (24) that when the
three collinear foci are all real the curve cuts the axis in four real
points, but when two of these foci are imaginary, the curve cuts
the axis in two real and two imaginary points.