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Analytical Geometry

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8 Jul 2009 19:47:33 IST

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when two circles are intersect each other,then the third circle which is passing through the two int

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when two circles are intersect each other,then the third circle which is passing through the two intersecting points.then is their any probability that the third circle is passing through the centers of the to circles.(if the two circles are same size or different size)

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NugoRama

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Joined: 11 Mar 2009

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8 Jul 2009 19:55:11 IST

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See ..if all the conditions given by you are fullfilled...then the figure wud be a hell lot symmetrical.

Hence by obserdvation i can say ..that it can happen only if they are of same size.

Another explanation ..by fixing 3 points ...we fix a circle.

two are already fixed ..the pts of intersection..

let third by the centre of one of the circles ..hence we have fixed the circle...

Now to make sure it also passes thru the centre of fourth ..another condition shud be given..which acc to my, by symmetry, is that both circles are of same size.

~validity of what i said ..just expired~

SAVVEJ

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Joined: 4 May 2007

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9 Jul 2009 10:32:08 IST

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let the two circles be :

S₁: x² + y² +2g₁x + 2f₁y + c₁ = 0

S₂ : x² + y² +2g₂x +2f₂y + c₂ = 0

now the equation of circle passing through their point of intersection is given by:

S₁ + S₂=0

now (-g₁ , -f₁) satifies this equation .also (-g₂,-f₂) satisfies it.

u will get on putting -g1,-f1 that :

similarly on putting -g2,-f2 u will get :

now both these values of lamda must be equal since they represent the same circle .

hence on equating we get the required condition as:

if u carefully observe the values of lamda u will find that for the first one it is ratio of radius² / (ength of tangent on S2 from -g1,-f1)²

and similarly u will find for the second one but sort of inverse of it.thus one of the cases when this will be true will be when the two circles cut each other orthoganally and have same radius.

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