562 
GEOMETRY. 
[790 
be divided at 0 in the given ratio, say we have OD — m, OF = em, where m is 
positive;—then the origin may be taken at 0, the axis Ox being in the direction OF 
(that is, from 0 to F), and the axis Oy at right angles to it. The distance of the 
point (x, y) from F is = V(# — em) 2 4- y 2 , its distance from the directrix is = x + m ; 
the equation therefore is 
(x — em) 2 + y 2 = e 2 (x + m) 2 ; 
or, what is the same thing, it is 
(1 — e 2 ) ot? — 2 me (1 + e) x + y 2 — 0. 
If e 2 = l, or, since e is taken to be positive, if e = 1, this is 
which is the parabola. 
y- — 4 mx = 0, 
If e 2 not =1, then the equation may be written 
(1-e 2 ) 
+ y 2 = 
m 2 e 2 (1 + e) 
1-e 
Supposing e positive and < 1, then, writing m — 
a ( 1 — e) 
e 
, the equation becomes 
that is, 
(1 — e 2 ) (x - a) 2 + y 2 = a 2 (1 — e 2 ), 
~ a) 2 , , . 
a 2 + a 2 (l- e 2 ) 
or, changing the origin and writing b 2 = a 2 (l— e 2 ), this is 
which is the ellipse. 
+ = \ 
a 2 + b 2 ’ 
And similarly if e be positive and >1, then writing m = 
becomes 
(1 — e 2 ) (x + a) 2 + y 2 = a 2 ( 1 — e 2 ), 
that is, 
O + a) 2 f , 
a 2 a 2 (1 - e 2 ) ' 
or changing the origin and writing b 2 = a 2 (e 2 — 1), this is 
which is the hyperbola. 
a (e — 1) 
x? y 2 _ 
a 2 ~b 2 = 1. 
, the equation 
18. The general equation ax 2 + 2hxy + by 2 + 2fy + 2gx + c = 0, or as it is written 
(a, b, c, f, g, h) (x, y, l) 2 = 0, may be such that the quadric function breaks up into 
factors, = (ax + /3y + 7) (oix + /3'y + 7'); and in this case the equation represents a pair 
of lines, or (it may be) two coincident lines. When it does not so break up, the 
function can be put in the form \ {(# — a') 2 + (y — b') 2 — e 2 (x cos a + y sin a — p) 2 ), or, 
equating the two expressions, there will be six equations for the determination of 
X, a', b', e, p, a ; and by what precedes, if a', b’, e, p, a are real, the curve is either