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

46 TANGENTIAL COORDINATES.
Upon this result Prof. Cayley has founded his theory of the
Absolute, which has been developed by Prof. Klein and others;
but the subject is beyond the scope of an elementary work1.
Foci.
75. We shall now explain how the circular points are employed to determine the foci of curves, and shall begin by proving
that:The lines joining the focus of a conic zith either of the circular
points at infinity touch the conic.
Let (a, 3) be the coordinates of the focus of the ellipse
x2/a2 + y2/b2 = 1.
The equation of the line joining (a, /3) to one of the circular points
is
x-a + (y - )=.................... (32).
Let
1 t: = _ ---,= a +................(33),
then if (32) is a tangent to the ellipse, 4 and V must be connected
by the equation
a22 + b22v= 1.
Substituting from (:3), we obtain
a +,3 = + (a2 - b2)2.
If a > b, the real values of a and / are given by
a= + (a2- b"2), =..................... (34),
whilst the imaginary ones are given by
a= 0, 3 = ~ (a2 - b2).
Equations (34) are the well known equations for determining
the real foci of the conic.
When the ellipse degenerates into a circle, a = b, and the two
real foci coincide with the centre, which is a double focus.
1 Cayley, "A sixth Memoir on Quantics," Math. Papers, Vol. n. p. 561. Klein,
Math. Annalen, Vol. xxxvII.; Lectures on Nicht-Euclidische Geometrie, Vol. I.
p. 61.