258 
[342 
342. 
ON THE CONICS WHICH PASS THROUGH THREE GIVEN 
POINTS AND TOUCH A GIVEN LINE. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. vi. (1864), 
pp. 24—30.] 
Consider the system of conics which pass through three given points and touch 
a given line; if among these we select the conics which touch an assumed line, it is 
easy to show analytically that there are four such conics, all real or else all imaginary; 
viz. the three points form a triangle, and if the two lines cut the three sides produced 
or cut the same two sides and the third side produced, then the conics are all real; 
but in every other case they are all imaginary. The latter part of the theorem may 
also be seen geometrically; in fact, if a triangle is inscribed in a conic, say first in 
an ellipse, or in a parabola, or in one branch of a hyperbola, then all the tangents 
of the conic (and therefore any two tangents whatever) cut the three sides produced, 
but if the triangle is inscribed in the two branches of a hyperbola (that is, two 
vertices on one branch and the remaining vertex on the other branch), then all the 
tangents of the conic (and therefore any two tangents whatever) cut the same two 
sides and the third side produced: and thus the only real conics are those wdiich cut 
the three sides produced, or else the same two sides and the third side produced. 
The analytical proof referred to is as follows: taking (x — 0, y = 0, z = 0) for the 
equations of the sides of the triangle, the equation of a conic through the three points is 
or, what is the same thing, 
t+ 9 - + h - =0, 
X y z 
2fyz + 2gzx + 2 hxy = 0, 
that is 
(0, 0, 0, f g, h#w, y, zf = 0.