Invert about the unit circle centered at the origin. $\omega$ turns into a horizontal line, and the hyperbola turns into the following: $$\begin{aligned} \frac{x^{2}}{\left(x^{2}+y^{2}\right)^{2}}-\frac{20y^{2}}{\left(x^{2}+y^{2}\right)^{2}}=24 & \Longrightarrow x^{2}-20y^{2}=24\left(x^{2}+y^{2}\right)^{2} \\ & \Longrightarrow 24x^{4}+\left(48y^{2}-1\right)x^{2}+24y^{4}+20y^{2}=0 \\ & \Longrightarrow\left(48y^{2}-1\right)^{2} \geq 4(24)\left(24y^{4}+20y^{2}\right) \\ & \Longrightarrow 1-96y^{2} \geq 1920y^{2} \\ & \Longrightarrow y \leq \sqrt{1/2016} \end{aligned}$$ This means that the horizontal line in question is $y=\sqrt{1/2016}$. This means that the diameter of the circle is the reciprocal of the distance between the point and line, which is $\sqrt{2016}$, so the radius is $\sqrt{504}$, and the answer is $504\pi$.