## Hyperbola with given asymptote

The following problem arose in the discussion about power generalization in PowerGeneralHyperbola.html .
Given a trapezium ABCD and a secant EE' of its parallels find a hyperbola passing through its vertices and
having one asymptote parallel to EE'.

According to the formula proved in the aforementioned reference, the intersection point P of the hyperbola
to construct will satisfy:
(ED*EC)/EP = (E'B*E'A)/E'P.                                 (*).
This, given the positions of {E,E'}, determines the position of P:
PE/PE' = (ED*EC)/(E'B*E'A).

All these hyperbolas are part of the pencil of conics passing through the four vertices of the trapezium.
The parallel sides are chords of the conic, hence the line MM' passing trhough their middles is a diameter
of the conic and passes through its center.
There is though a little problem here. The above condition is also a necessary condition in certain cases
to pass through these four points also for a parabola. In fact in PowerGeneralParabola.html was shown that for
all points P on the parabola passing through {A,B,C,D} (see ParabolaCircumscribingTrapezium.html on the
construction of the unique parabola with this property) the ratio  r=(ED*EC)/EP is independent of the position
of P when EE' is parallel to the axis of the parabola i.e. parallel to MM'.

More general, parallel directions EE' determine different P's but lying on the same hyperbola. This follows
from the fact that then points P satisfying the equality (*) determine a conic. This is easily seen by taking
for x-axis DC and y-axis the line MM' and translating the equation in terms of (x,y).
Thus all these hyperbolas are parameterized by the location of point E on the half-line M'C by joining it to
F and determining the direction of the asymptote through FE.

CircumconicsTrapezium.html displays various members of the pencil of all conics circumscribing a trapezium.
There it is seen that the part of the pencil consisting of hyperbolas has the corresponding centers of the
hyperbolas on the half-line SO, where S is the middle of MM'. The opposite half contains the centers of the
remaining members which are all ellipses. The middle S of MM' is the "center" of the degenerate parabola
consisting of the two parallel lines {AB, CD}.