THE CONICS, BOOK I
137
pai’abola parallel to AC] in the case of the hyperbola it meets
the other half of the double cone in P'; and in the case of the
ellipse it meets the cone itself again in P'. We draw, in
the cases of the hyperbola and ellipse, AF parallel to PM
to meet BG or BG produced in F.
Apollonius expresses the properties of the three curves by
means of a certain straight line PL drawn at right angles
to PM in the plane of the section.
In the case of the parabola, PL is taken such that
PL: PA = BG 2 :BA.AG\
and in the case of the hyperbola and ellipse such that
PL : PP' = BF. FG: A F\.
In the latter two cases we join P'L, and then draw VR
parallel to PL to meet P'L, produced if necessary, in R.
If UK be drawn through V parallel to BG and meeting
AB, AG in H, K respectively, HK is the diameter of the circular
section of the cone made by a plane parallel to the base.
Therefore QV 2 = HV. VK.
Then (1) for the parabola we have, by parallels and similar
triangles,
HV-.PV = BG-.CA,