Problem : If the distances from origin to the centre of three circles $x^2+y^2+2\lambda_ix-c^2=0$ (i=1,2,3) are in G.P ( Geometric progression). Then lengths of tangents drawn to them from any point on the circle $x^2+y^2=c^2$ are in
Solution : Here centre of the first circle is $(-\lambda_1,0)$...(i) ;
centre of the second circle is $(-\lambda_2,0)$...(ii) ; centre of the third circle is $(-\lambda_3,0)..(iii)$
Distance from the origin from these centres (i), (ii) \& (iii) are :
$(\lambda_1, \lambda_2,\lambda_3)$ respectively. As per the given condition : $\lambda_2^2 = \lambda_1\lambda_3$ ( as they are in G.P )
Let the point on the circle $x^2+y^2=c^2$ is $(\alpha, \beta)$ Now the distance from this point ie. $(\alpha,\beta)$ from the given three circles are given by :
Please suggest how to proceed on this further I am not getting any clue.. Thanks.