120 
CONIC SECTIONS 
Proof from Pappus. 
The proof in the case where the given ratio is different from 
unity is shortly as follows. 
Let S be the fixed point, SX the perpendicular from S on 
the fixed line. Let P be any point on the locus and PX 
k an sk' 
perpendicular to SX, so that SP is to NX in the given 
ratio (e) ; 
thus _ (p^2 + £jy2j . NX\ 
Take K on SX such that 
e 2 = 81y 2 : NIP ; 
then, if K' be another point on SN, produced if necessary, 
such that NK = NK', 
e 2 :1 = (PN 2 + SN 2 ) : NX 2 = SN 2 : NK 2 
= PN 2 : (NX 2 — NK 2 ) 
= PN 2 : XK. XK'. 
The positions of N, K, K' change with the position of P. 
If A, A' be the points on which N falls when K, K' coincide 
with-X respectively, we have 
SA : AX = SN: NK = e : 1 = SN: NK'= SA'-.A'X. 
Therefore SX : SA = SK : SN = (1 +e):e, 
whence ( 1 + e) : e = (SX - SK) : (SA - SN)