A tangent at a point on the circle $x^{2} + y^{2} = a^{2}$ intersects a concentric circle $C$ at two

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A tangent at a point on the circle $x^{2} + y^{2} = a^{2}$ intersects a concentric circle $C$ at two points $P$ and $Q$. The tangents to the circle $C$ at $P$ and $Q$ meet at a point on the circle $x^{2} + y^{2} = b^{2}$ then the equation of circle ' $C$ ' is:

Accepted Answer

[Key Concept: A tangent to one circle that meets a concentric circle subtends a right angle / fixed chord — use the radii.]Equation of $AB$ is $xh + yk = r^{2}AB$ is tangent to $x^{2} + y^{2} = a^{2}$$${\Rightarrow \frac{r^{2}}{\sqrt{h^{2} + k^{2}}} = a }{\Rightarrow r^{2} = ab}$$$$\left\lbrack \because h^{2} + k^{2} = b^{2} \right\rbrack$$⇒ Equation of circle is $x^{2} + y^{2} = ab$

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