Question

Tangents are drawn from the point on the circle $x^2+y^2=a^2$to the circle $x^2+y^2=b^2$ and the chord of contact touch the circle $x^2+y^2=c^2$ then $a,b,c$ will be in

• A. $AP$
• B. $GP$
• C. $HP$
• D. $None$

Answer 1

The correct option is B $GP$

let the point on the circle $x^2+y^2=a^2$ be $(x_1,y_1)$

eq'n of chord of contact from a point on a circle is $x{x}_1+y{y}_1=r^2=b^2$

Tangent to any circle with center origin is of the form, $y=mx+r\sqrt{1+m^2}$

Tangent to the circle $x^2+y^2=c^2$ is of the form, $y=mx+c\sqrt{1+m^2}$

given chord of contact is the tangent to the circle $x^2+y^2=c^2$

So, $m=\frac{-x_1}{y_1},c\sqrt{1+m^2}=\frac{b^2}{y_1}$

$c^2(1+m^2)=\frac{b^4}{y_1^2}$

$c^2(1+\frac{x_1^2}{y_1^2})=\frac{b^4}{y_1^2}$

$c^2(x_1^2+y_1^2)=b^4$

$c^2{a}^2=b^4$

$ca=b^2$

a,b,c are in GP