Question

If the squares of the lengths of the tangents from a point $P$ to the circles $x^2+y^2=a^2,x^2+y^2=b^2$ and $x^2+y^2=c^2$ are in $A.P$, then $a^2,b^2$ , $c^2$ are in

• A. A.P.
• B. G.P.
• C. H.P.
• D. A.G.P

Answer 1

The correct option is A A.P.

Let P$(h,k)$ , than length of tangent from P to circle is
$x^2+y^2=a^2$
$L_1=\sqrt{h^2+k^2-a^2}$
and $L_2=\sqrt{h^2+k^2-b^2}$
$L_3=\sqrt{h^2+k^2-c^2}$
Given $L_1,L_2,L_3$ are is A.P.
$2(h^2+k^2-b^2)=h^2+k^2-a^2+b^2+k^2-c^2$
$2b^2=a^2+c^2$
So, $a^2,b^2,c^2$ are also in A.P.