Question

If the distance from the origin for the centres of three circles $x^2+y^2+2{\lambda }_{i}x-c^2=0(i=1,2,3)$ are in G.P., then the lengths of the tangents drawn to them from any point on the circle $x^2+y^2=c^2$ are in

• A. A.P.
• B. G.P.
• C. H.P.
• D. None of these

Answer 1

The correct option is B G.P.

Given ${{\lambda }_2}^2={\lambda }_1{\lambda }_2$
Any point on $x^2+y^2=c^2$ is $(c\cos \theta ,c\sin \theta )$
${t_1}^2=-2{\lambda }_{1}c\cos \theta ,{t_2}^2=-2{\lambda }_{2}c\cos \theta ,{t_3}^2=-2{\lambda }_{3}c\cos \theta$
Clearly, ${\left({t_2}^2\right)}^2={t_1}^2.{t_3}^2(\because {{\lambda }_2}^2={\lambda }_1{\lambda }_2)$
$\therefore {t_1}^2,{t_2}^2,{t_3}^2$ are in G.P hence $t_1,t_2,t_3$ are also in G.P.