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.