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

EQUATION OF TANGENTS FROM A POINT.

51

project into real points. Hence the existence of imaginary foci
must not be overlooked, otherwise we should lose sight of various
properties connected with the points of intersection of tangents
drawn to a curve from a pair of real nodes or cusps.
82. We shall conclude this chapter with two miscellaneous
propositions.
To find the equation of the tangents drawn from the point (h, k)
to a curve.
Let
+ = 1...........................(38)
be any tangent; and
F (:, r)= 0...........................(39)
the tangential equation of the curve. Since (38) passes through
(I', k),
h k + r = 1,
whence by (38),
f (kx - hy) = k - y,
r (kx - hy) = x- h,
whence the equation of the tangents is
Fkx -hy ' kx- hy}
83. A straight line is drawn through a fixed point 0; to find
the locus of the points of intersection of the tangents at the points
where it cuts the curve.
Let U= 0 be the Cartesian equation of the curve referred to
0 as origin; and let V=0 be the first polar of any point (h, k).
Transform to polar coordinates and eliminate r; then the resulting
equation will determine tan 0, where 0 is the vectorial angle
of the point of contact of any tangent drawn from (h, k). The
degree of this equation is necessarily the same as the class of the
curve.
Let (h, k) be the point of intersection of the pair of tangents
at any two points P and Q where a straight line through 0 cuts
the curve; then since tan 0 = tan (nr + 0) two of the roots of the
equation for tan 0 must be equal; whence the discriminant of this
equation equated to zero is the required locus.
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