392] 
AND TOUCH TWO GIVEN LINES. 
49 
Chord of Contact. 
OP 
OL 0 = 0). 
00. 
OP,. 
OX {« + y V {run) = 0}. 
OG (parallel to y = 0). 
OH (parallel to x = 0) and so back to 
OP,. 
Centre. 
P,, at infinity on hyperbola. 
L, {z = 0, « — y = 0). 
©, the line joining this with 0 being always 
behind 0©. 
P i} at infinity on hyperbola. 
X {x = 0, y = 0). 
G (on line y = 0). 
H (on line x = 0) and so on to 
P 2 . 
I have treated separately the case f (mn) = 1. 
Consider the conics which touch the lines y — x = 0, y + x = 0 and pass through 
the points 
[x = l, y = f (1 - c 2 )}, \x=l, y = - V (1 - c 2 )}. 
The equation is of the form 
and to determine Je, we have 
y 2 — x 2 + k{x — a) 2 = 0, 
1 — c 2 — 1 + k (1 — a) 2 = 0, and therefore k = - 
c 
The equation thus becomes 
(1 — a) 2 (y 2 — x-) + c 2 {x — a) 2 = 0, 
that is 
(1 — a) 2 y 2 + {c 2 — (1 — a) 2 } x 2 — 2c 2 ax + c 2 a 2 = 0, 
c-a 
= 0. 
or as this may be written 
(1 - a) 2 y 2 + [c 2 - (1 - a) 2 ] - c ,_({_ a y 
Hence the nature of the conic depends on the sign of c 2 —(1 —a) 2 , viz. if this be 
positive, or a between the limits 1 + c, 1 — c, the curve is an ellipse, 
c 2 « 2 (1 — of 
c 2 — (1 — a) 2 
«-coordinate of centre = 
c-a 
which is positive, 
«-semi-axis 
?/-semi-axis 
c 2 - (1 - a) 2 ’ 
+ ca (1 — a) 
c 2 - (1 - a) 2 ’ 
COL 
V{c 2 -(l-a) 2 } 
The coordinate of centre for a = l + c is = + oo (the curve being in this case a parabola 
P,) and for a = 1 - c it is also = + oo (the curve being in this case a parabola P,). The 
coordinate has a minimum value corresponding to a — V (1 — c 2 ), viz. this is = \ {1 + V (1 — c 2 )}. 
7 
c. VI.