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Q:-If the distances between the origin and centers of the three circles x 2 +y 2 - 2λ r x -a 2 =0 ; r=1,2,3 are in G.P then the lengths of tangents to the circles from any point on the circle x 2 +y 2 =a 2 are in__________ 1.A.P 2.G.P 3.H.P 4.A.G.P 5.None of these

Q:-If the distances between the origin and centers of the three circles x2+y2 - 2λr x -a2=0 ; r=1,2,3 are in G.P then the lengths of tangents to the circles from any point on the circle x2+y2=a2 are in__________


1.A.P


2.G.P


3.H.P


4.A.G.P


5.None of these

Grade:12

1 Answers

Badiuddin askIITians.ismu Expert
147 Points
11 years ago

Dear Tapasranjan

x2+y2 - 2λr x -a2=0

(x-λr)2 +y2 =a2 + λr2

 center of circle( λr,0)

   so distance of center from the origin. = λr

   so λ1 2 ,λare in GP

now let a general point (acos θ ,asinθ) on circle x2+y2=a2    

so length of tangent from this point to the given circle

       = [(acos θ)2+(asinθ)2 - 2λacos θ -a2]1/2

       =[ - 2λacos θ ]1/2

 so given λ1 2 ,λare in GP

   so   -1 acos θ,-2λ2 acos θ,-3 acos θ  will also be in GP

         [ - 2λ1 acos θ ]1/2 , [ - 2λ2 acos θ ]1/2  , [ - 2λ3 acos θ ]1/2  will also be in GP

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Badiuddin


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