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

190

SPECIAL QUARTICS.

286. Tangents at the extremities of a chord through the external focus subtend equal angles at the internal focus; also the
locus of their point of intersection is a cissoid.
The first part follows at once, since
TFPP =TQP= TPQ = TFQ.
To prove the second part, let TFF -=x; then, since OT is
perpendicular to PQ,
= -27r-OFT='FOT= 08;
whence, if (x, y) be the coordinates of T,
x2 + y" 2
Ya~ = cosec2 = seec 1 -........ (9)
Y2 / 2 I os1 + cos 0...
Let M1 be the middle point of PQ, then
OM sin 0, + F1M cos 01 =f= (a2 - b2)/2b;
also, by (4), F M = (a2 cos 01 + b2)/2b,
a2 sin2 0, - b (1 + cos 0i)
whence OM = sin b
2b sin 80
Also OF =ftan 01 - OM sec 08
b(1 + cos 8)
2 sin 0
and = OF sin 0 = b (1 + cos 0)...............(10),
whence by (9) and (10), the locus of T is the cissoid
x (x2 + y2)= by 2.......................(11).
287. The locus of the point of intersection of two tangents at
the extremities of a chord through the node is a nodal circular
cubic.
The equation of the limacon in Cartesian coordinates is
(2 +2 + bx)2= a (2 + y2)......... (12).
Let (h, k) be any point; write down its polar cubic, transform
to polar coordinates and then eliminate r by means of the polar
equation of the curve, and we shall obtain
{(a2 - bh) tan2 0 + 2bk tan 0 + a2 + b2 + bh}2
= a2 (k tan 0 + 2b + h)2 (1 + tan2 0),