574 
PROBLEMS AND SOLUTIONS. 
[485 
a fixed line through 0 meeting the second conic in the points A and B\ then con 
sidering the conic which passes through P and touches at I, J the lines AI, AJ 
respectively, and also the conic which passes through P and touches at I, J the lines 
BI, BJ respectively; the envelope of the lines which cut harmonically the last-mentioned 
two conics is a conic independent of the position of P. 
2. Taking x = 0, y = 0, z — 0 for the equations of the lines 01, JI, and OJ 
respectively, the equations of the two given conics are 
xz — y- = 0, kxz — y n - = 0 ; 
hence the coordinates of P may be taken to be 
x : y : z = 1 : 9 : 6-, 
and the coordinates of the points A and B may be taken to be 
x : y : z = 1 : ka : ka 2 , and x : y : z = 1 : — ka : ka 2 . 
The equations of the lines AI, AJ are 
kax — y = 0, z — ay = 0 ; 
hence the equation of the conic touching these lines at the points I, J respectively, 
and also passing through the point P, is 
(kax-y) (z-ay) _ y-_ 
(ka — 6) (6 — a) 6 
and similarly the equations of the lines BI, BJ being 
koLx + y = 0, z+ ay = 0, 
the equation of the conic touching these lines at the points I, J respectively, and 
also passing through the point P, is 
(fox + y) (z + ay) y- 
(ka + 0) (0 + a) 0’ 
or multiplying out and reducing, if the equations of the two conics are represented by 
(a, b, c, f, g, K$x, y, zf = 0, (a', b', c, /', g, K\x, y, zf = 0, 
respectively, then the values of the coefficients are 
CL — 0, 
5 = 2 (ka + 0- — ka0), 
b' = 2 (— ka 2 — 6- — ka6), 
a' — 0, 
c = 0, 
f=-e, 
g = 0ka, 
h = — 6Ica 2 ,