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

80

SPECIAL CUBICS.

whence by the equation of the curve
Z az2= - a/b...........................( ).
Let POB = 0, pOB = 9'; then z1 = tan 0, z = - tan 0', whence
tan0- tan 0' - 2k/h)
tan 0 tan 9' =a/b j '........ (12).
Accordingly
tan 0 + tan ' = 2 (bk2 + ah2) /hb = 2X (say).
Produce the ordinate at p to meet the curve in P', then
P'OB = pOB = '; and the equation of the two lines OP, OP' is
ax2 - 2bXxy + by2 =.................(13).
The equation of the curve is
x (x2 + y2)- ax 2 by2 = 0..................(14).
Adding (13) and (14) we get
2 + y2 _ 2bXy =....................(15),
which is the equation of the circle circumscribing the triangle
OPP'.
Multiply (15) by a and subtract from (13) and we get
(b - a) y + 2bX (a - x) = 0.
This is the equation of the straight line which passes through
P and P', and since it is satisfied by y = 0, x = a, it passes through
the vertex A.
132. The circle circumscribing the triangle OPp passes through
a fixed point on the axis.
From (12) it follows that the equation of OP, Op is
ax2 - by2 - 2bkxy/h = 0.
Subtracting this from (14) we obtain
X2 + y2 _ 2ax + 2bky/h = 0,
which is the equation of the circle which passes through OPp.
This obviously passes through the point x = 2a, y = 0.