PROBLEMS AND SOLUTIONS. 
603 
[485 
485] PROBLEMS AND SOLUTIONS. 603 
>} = o, 
which is apparently a quartic curve; but it is obvious a priori that the locus includes 
as part of itself the line AB which joins the two given points. In fact, there is in 
the series of confocal conics one conic which touches the line in question, and since 
for this conic one of the tangents from A and also one of the tangents from B is 
four distinct 
the line AB, we see that every point of the line AB belongs to the required locus. 
The locus is thus made up of the line in question and of the cubic curve. 
ve. In fact, 
jersection of 
ugh P, the 
that at P 
locus of P 
ed if it be 
n A, B to 
,rk that the 
6. To effect the reduction it will be convenient to write ax, by in the place of 
x, y, {axo, by 0 , ax 1} by x in place of x 0 , y,, x 1 , y 1 ,) and thus consider the equation under 
the form 
a 2 (x - x 0 ) 2 + b 2 (y - y 0 ) 2 __ a 2 (x - x x ) 2 + b 2 (y - y^ 2 
(xy 0 - x 0 y) 2 -{x- x 0 ) 2 -(y - y,J (xy x - xy) 2 -{x- x{f -{y- y,) 2 ’ 
it is to be shown that this equation represents the line L = 0, and a cubic curve. 
Writing for a moment x 0 = x+%,, y 0 = y + V() , and x 1 = x + ^ 1 , y 1 = y + Vl , the equation 
given conic 
becomes 
a% 2 + b 2 tj 0 2 a% 2 + b‘% 2 
take to be 
the conic 
( x Vo - y%of - £o 2 - Vi, 2 (%vi - yh) 2 ~ ~ Vi ’ 
and hence, multiplying out, the equation is at once seen to contain the factor 
3 focal conic 
ZoVi— %iVo (which is in fact the determinant just mentioned), and when divested of 
this factor the equation is 
hose of B ; 
a 2 [{x 2 -l){^Vi+ ZiVo) ~ = b 2 [(;y 2 -1) {^ oVl + £ lVo ) - 2xy VoVl ]. 
Writing herein for % 0 , y 0 , y x their values, and consequently 
¿Toll = X 2 — X(X 0 + Xj) + x 0 x Y , 
VoVi = y 2 - y (2/o + 2/1) + y*yi, 
£oVi + ZiVo = 2xy-x {y, + yi)-y (x 0 + x x ) + xy 1 + xy 0 , 
and arranging the terms, the equation is found to be 
(a 2 x 2 + b 2 y 2 ) [- x (y x +y 0 )-y (ah + a?„)] + (a*«* + by) (x 0 y 1 + xy,) - 2xy [a? (1 + xyvj -b 2 { 1 + y 0 y,)] 
+ (a 2 - b 2 ) [x {y 1 + y 0 ) + y(x x + x 0 ) - (x 0 y 1 + xy,)] = 0, 
which is the required cubic curve. 
3ver. 
7. Restoring the original coordinates, or writing -, j, —, &c. in place of x, y, x„ &c., 
CL 0 CL 
we have 
(x 2 + y 2 ) [- x (y l + 2/0) + y («1 + «0)] + (P ~ y 2 ) (^o2/i + xy,) - 2xy (a 2 - b 2 + x 0 x 1 - y,yi) 
+ (a 2 - b 2 ) [x {y 1 +y 0 ) + y {x 1 + x,) - (x,y 1 + xy,)] = 0, 
s 
? 
which is a circular cubic the locus of the intersections of the tangents from the 
arbitrary points {x,, y 0 ), {x li y x ) to the series of confocal conics ^ — ^ ^ = 1; the 
origin of the coordinates is at the centre of the conics. 
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