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Solution

Equation of chord of contact from $(a\cos\theta, a\sin\theta)$ to circle $x^2 + y^2 = b^2$ is $a(\cos\theta \,x + \sin\theta \,y) = b^2.$ For this line to be tangent to $x^2 + y^2 = c^2$, the perpendicular distance from origin = radius of that circle. $\Rightarrow \dfrac{|b^2|}{\sqrt{a^2}} = c$ $\Rightarrow b^2 = a c$ Hence, $a, b, c$ are in G.P.