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Here's the equation I need to solve (for $x$)...

$$r\sqrt{x^{2}-r^{2}}+\left(\frac{1}{b}-r^{2}\right)\left(\arccos\left(\frac{r}{x}\right)\right)=\frac{\pi}{2b}$$

is there a solution in terms of the constants $r$ and $b$?

Hmm...

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    $\begingroup$ Solve $\large \sqrt{\vphantom{\Large A}x^{2} - 1\,} + \mu\,\arccos\left(1 \over x\right) = \nu$ first. $\large\mu \equiv {1 \over br^{2}} - 1$ and $\large\nu \equiv {\pi \over 2br^{2}}$. $\endgroup$ – Felix Marin Sep 27 '13 at 0:05
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Start by writing $$x=r\sec t$$ This eliminates the square root and the inverse cosine in terms of $\tan t$ and t.

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