Q:-If the distances between the origin and centers of the three circles x²+y² - 2λ_r 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

Dear Tapasranjan

x²+y² - 2λ_r x -a²=0

(x-λ_r)² +y² =a² + λ_r²

 center of circle( λ_r,0)

   so distance of center from the origin. = λ_r

   so λ₁ ,λ_{2 ,}λ₃  are 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 - 2λ_r acos θ -a²]^1/2

       =[ - 2λ_r acos θ ]^1/2

 so given λ₁ ,λ_{2 ,}λ₃  are in GP

   so   -2λ₁ acos θ,-2λ₂ acos θ,-2λ₃ 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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