[alogo] 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.

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[1_0] [1_1] [1_2] [1_3]

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

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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.

[alogo] 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.

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

[alogo] 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'.

[0_0] [0_1] [0_2]

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:


[alogo] 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.

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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.

[alogo] 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 LA. 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 LA 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.

See Also



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

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