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. $a^2,b^2,c^2$ are in A.P
• B. $\frac{1}{a^2},\frac{1}{b^2},\frac{1}{c^2}$ are in H.P
• C. $a^2,b^2,c^2$ are in G.P
• D. $\frac{1}{a^2},\frac{1}{b^2},\frac{1}{c^2}$ are in A.P

Answer 1

Answer: (A), (B)

The correct options are
A $a^2,b^2,c^2$ are in A.P
B $\frac{1}{a^2},\frac{1}{b^2},\frac{1}{c^2}$ are in H.P

Let $P(x,y)$ be the point.

Length of the tangent from $P(x,y)$ to $x^2+y^2=r^2$ is $=\sqrt{x^2+y^2-r^2}$

From the given condition we get,

$2(x^2+y^2-b^2)=x^2+y^2-a^2+x^2+y^2-c^2$
$\Rightarrow {2b}^2=a^2+c^2\Rightarrow a^2{b}^2{c}^2$ are in A.P
So $\frac{1}{a^2},\frac{1}{b^2},\frac{1}{b^2}$ are in H.P