1902-3.] Equation of a Pair of Tangents to a Conic. 
413 
On the Equation of a Pair of Tangents to a Conic. By 
Professor A. H. Anglin, Queen’s College, Cork. 
(Read February 16, 1902.) 
{Abstract.) 
The equation to the pair of tangents from the point ( x , y) to 
the conic <£(x, y) = 0 is usually obtained in the form 
<K%, V ) "<f>( x ’ ?/') = T2 
where T = 0 is the equation to the chord of contact. This equation 
admits of reduction ; and we propose to obtain the reduced form 
independently, and to supply its geometrical interpretation. 
Taking the case where the equation to the conic is 
ax 2 + 2 hxy + by 2 = 1 , 
the line lx 4- my + n — 0 will be a tangent if 
am 2 - 2 him + bl 2 = {ab - h 2 )n 2 . 
If the line pass through (x, y) we have 
V(y - v) = ~ x ) = n l( x y' - x ’y)- 
Hence, for any point on either tangent from (x\ y) to the conic, 
substituting for Z, m, n in the above relation we get 
a{x - x) 2 + 2 h(x - x')(y - y) + b(y - y) 2 = (ab - h 2 )(xy — xy) 2 , 
the required equation to the pair of tangents. 
If Q be the point (x, y'), P any point ( x , y) on either tangent, 
and C the centre of the conic, this equation is the result of 
equating two expressions which are equal to the square of 
2 A CPQ. 
If the case of the most general form of equation 
ax 2 + 2 hxy + by 2 + 2 gx + 2 fy + c = 0 
be deduced from the preceding, we shall get 
a(x - x) 2 + 2 h(x - x')(y - y') + b(y - if) 2 + 
(ab - h 2 ) 2 
x y 1 
x y 1 
x y" 1 
0 , 
where (x , y") is the centre, and A the discriminant of the conic ; 
and from which the geometrical interpretation is obvious. 
