389] 
ON A LOCUS DERIVED FROM TWO CONICS. 
31 
I suppose in particular that the two conics are 
x 2 + my 2 — 1 = 0, 
mx 2 + y- — 0, 
the equation of the quartic is 
4 (2k + l) 2 m 2 (a? + my 2 - 1) (mx- + y 2 - 1) - {(m 2 + m) (x 2 + y 2 ) - m 2 - l} 2 = 0 ; 
,. ^ (m +1) 2 . . 
or putting A = 4 ( 2k + 1 ^ 2 , this is 
\ lx 2 + y 2 — 
m 2 +1 \ 2 
m 2 -f m 
— (x 2 + my 2 — 1) (mx 2 + y 2 — 1) = 0. 
To fix the ideas, suppose that m is positive and > 1, so that each of the conics 
is an ellipse, the major semi-axis being = 1, and the minor semi-axis being = . 
y(m) 
For any real value of k the coefficient A is positive, and it may accordingly be assumed 
that A is positive. 
We have ■. x > — <1, or the radius of the circle is intermediate between 
m (m + 1) m 
the semi-axes of the ellipses, hence the points of contact on each ellipse are real points. 
Writing for shortness 
the equation is 
a = 
m 2 + 1 
m 2 + m ’ 
(x 2 + my 2 — 1) (mx 2 + y 2 — 1) — A (x 2 + y 2 — ct) 2 = 0. 
For the points on the axis of x, we have 
(oc 2 — 1) (mx 2 — 1) — A (x 2 — a) 2 = 0, 
that is 
and thence 
(m — A) x* + {— (1 + m) + 2Aoc} x 2 + (1 — Aa 2 ) = 0, 
(m - A) x 2 = \ (i + m) - Aa ± 1 V {(m - l) 2 + 4A (1 - a) (1 - ma)}, 
or, substituting for a its value, this is 
1 
X i 71% “j - j i \ 
(m -X)x 2 = \ (m + 1) m + i m ± + ^ ~ 
Remarking that the values m, K^+l) 2 are in the order of increasing magnitude, 
W-i 
ml