Question

The chord of contact of tangents drawn from any point on the circle $x^2+y^2=a^2$ to the circle $x^2+y^2=b^2$ touches $x^2+y^2=c^2$, where $a,b,c>0,$ then $a,b,c$ are in

• A. $2b=a+c$
• B. $b=\sqrt{ac}$
• C. $b=\frac{2ac}{a+c}$
• D. $b^2=a^2+c^2$

Answer 1

The correct option is B $b=\sqrt{ac}$

Equation of chord of contact drawn from the point $(x_1,y_1)$ on $x^2+y^2=a^2$ to the circle $x^2+y^2=b^2$ will be given by
$x{x}_1+y{y}_1=b^2$
Also ${x_1}^2+{y_1}^2=a^2$
As we know that it touches the circle
$x^2+y^2=c^2$
So, the distance from the origin will be,
$|-\frac{b^2}{\sqrt{x_1^2+y_1^2}}|=c\Rightarrow b^2=ac$

Hence, $a,b,c$ are in $G.P.$