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

FOCAL CONICS AND FOCI.

151

and therefore the required locus is
{(12 + m2 _ n2) r2- 2m2ax + 2n2 (ex +fy) + m2a2 - n2 (e2 +f2)}2
= 412lm2r2 (r2 - 2ax + a2)2..................(34),
which is the equation of a bicircular quartic. To prove that the
three points are foci, we shall show that the line x + ty = 0 is a
tangent to the quartic. Substituting tx for y in (34) it becomes
2n2 (e + f ) x - 2m2ax + n 2a2- n2 (e2 +f2)}2 = 0,
which is a perfect square, and therefore shows that the line
x + ty = 0 touches the curve at the imaginary point determined
by this equation.
222. It follows from ~ 202 and 81 that if a conic be inverted
from any point 0, the point 0 and the two inverse points of the
foci of the conic are foci of the quartic. We shall now prove
geometrically that if P be any point on the quartic OP, SP and
HP are connected by a linear relation.
Let C', S', H' denote the centre and foci of the conic, P' any
point on the conic, 2a its lnajor axis; then
H'P' OP' S' P' OP'
HP OH; SP OS
also since S'P' + H'P' = 2a, we obtain
2a HP SP
OP = OH +.OS( ********-*^(35),
or if k is the radius of inversion
2aOP/k2 = HPOH + SP/OS...............(36),
which is the required linear relation.
Let 0 coincide with the centre C of the conic; then
OH'= OS'= ae,
whence (35) becomes
e (SP + HP)= CP.
When the conic is a hyperbola, this becomes
e (SP - HP)= CP,
also SP2 + HP2 = 2 (CP2 + CS2),
whence SP. HP = CP2 (1 - 2/e2) + CS2.