if the distances from the origin of the centres of 

three circles  $x^2+y^2+2{\lambda }_{i}x-c^2=0(i=1,2,3)$  are in GP,

 then the lengths of the tangents drawn to them from any point on the circle 

 $x^2+y^2=c^2$ are in 

detailed solution

Correct option is B

The centres of the given circles are −λi,0(i=1,2,3)  The distances from the origin of the centres are λ1,λ2  and λ3.  It is given that  λ22=λ1λ3Let P(h,k) be any point on the circle x2+y2=c .Then h2+k2=c2Now,   Li=  Length of the tangent from (h,k) to x2+y2+2λixx−c2=0⇒ Li=h2+k2+2λih−c2⇒Li=c2+2λih−c2                                                            ∵h2+k2=c2⇒Li=2λjh,    i=1,2,3. ∴   L22=2λ2h⇒ L22=2hλ1λ3⇒ L22=2λ1h2λ3h=L1L3                                                                                 ∵λ22=λ1λ3Hence, L1,L2,L3 are in G.P