Question

If the distance from the origin of the centers of the three circles $x^2+y^2+2a_{i}x=a^2(i=1,2,3)$ are in G.P., then the length of the tangent drawn to them from any point on the circle $x^2+y^2=a^2$ are in

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

Answer 1

The correct option is B G.P.

The centers of the three given circles are $(-{\alpha }_1,0),(-{\alpha }_2,0)$ and $(-{\alpha }_3,0)$.

the distance of the three points from the origin are ${\alpha }_1,{\alpha }_2$ and ${\alpha }_3$.

Given: ${\alpha }_1,{\alpha }_2$ and ${\alpha }_3$ are in G.P.

$\Rightarrow {{\alpha }_2}^2={\alpha }_1{\alpha }_2$

Now, coordinate of any point on the circle $x^2+y^2=a^2$ are $(a\cos \theta ,a\sin \theta )$.

$\therefore$ The lengths of the tangents drawn from the point $(a\cos \theta ,a\sin \theta )$ to the three given circles are

$\sqrt{2{\alpha }_{1}a\cos \theta },\sqrt{2{\alpha }_{2}a\cos \theta }$ and $\sqrt{2{\alpha }_{3}a\cos \theta }$

using (1) are in G.P.