PowerGeneral

1. Power generalization

Fix two directions AB, CD. From an arbitrary point P draw two parallels to these directions
intersecting a conic at points {A',B'} and {C',D'} correspondingly. Then the ratio of the products k =
(PA'*PB')/(PC'*PD') is independent of the position of point P.

The proof (taken from [Loney, p. 371]) is a calculation using the general equation of the conic.

Obviously last expression is independent from the position (x',y') of P.

Remark-1 By setting P at the center of the conic (if any), follows, that its value equals the ratio of the
squares of the diameters which are parallel to AB and CD.
Remark-2 The properties are reminiscent of the properties of power of a point with respect to a
circle, therefore the name on the title of this section. In fact, if the conic is a circle then k=1 for all
possible directions. The inverse is also true: If k=1 for all possible directions then the conic is a
circle. It even suffices to check if k=1 for three pairs of three different directions.
Remark-3 The proposition has another famous consequence studied in CarnotConics.html .
Remark-4 The discussion is applicable in all these cases in which the two secants parallel to
{AB,CD} both intersect the conic at two real points. The cases for which one of the secants intersects the
conic at one only point (can occure in parabolas and/or hyperbolas) is discussed in the files
PowerGeneralParabola.html and PowerGeneralHyperbola.html respectively.

2. Tangents length-ratio

The ratio of lengths of the tangents PA/PB from P to a conic (c) is equal to the ratio A'A''/B'B'' of the
diameters which are parallel to these tangents.

Follows from the previous discussion and remark by taking the special position of P, so that the
secants become tangents.

3. Parallel secants

Fix a direction AB and consider the intersections with a conic (c) of the parallels to AB from P and
P'. Then the ratio
k = (PA₁*PB₁)/(P'A₂*P'B₂) is independent of the direction AB, and depends only from the position
of P and P'.

The proof follows from the initial calculation by dividing the corresponding ratios for the two points
P(x',y'), P'(x'',y''), the polar angle being the same:

4. Circle intersections

Let circle (c) intersect the conic (c') at four points {A,B,C,D}. Then lines AB and CD are equal
inclined to the axes of the conic.

By the initial proposition (EA*EB)/(EC*ED) is independent from the position of E. Since the four
points are concyclic, this ratio is 1. Thus, by selecting another E so that the parallels from E to CD
and AB become tangents, we see that these two tangents must have equal lengths. This is only
possible if that E is on the axis of the conic.

5. Conic and Circle-bundle intersections

Let {A,B} be two points on the conic (c) and consider the bundle of circles passing through these
two points. Then the other intersection points {C,D} of the circles define lines which are parallel to
a fixed direction.

This is immediate consequence of the previous proposition. Lines DC and AB are equal inclined to
the axes of the conic.
Remark-1 Letting A converge to B we arrive at the following result. Consider the circle bundle of
circles tangent to a conic at a fixed point A of the conic with tangent there L_A. The two other
intersection points {C,D} of the bundle members and the conic define lines CD that are parallel to
direction L' which is symmetric to L_A with respect to an axis of the conic.
Remark-2 The proposition is in general not true if the circle bundle is of non-intersecting type or has
the base points {A,B} not lying on the conic. There are though some cases in which the circles of a
bundle of non-intersecting type intersect a hyperbola along points in parallel directions (see
CircleBundleTransformationHyperbola2.html ).
In the file RectHypeParaChords.html is contained a synthetic proof of this fact for the case of
rectangular hyperbolas.

Bibliography

[Loney] Loney, S. L. Coordinate Geometry. Delhi, AITBS Publishers, 1991

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