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Calculus Level 4

Suppose we have two concentric circles of radii \(3\) and \(4\), respectively. Let \(R\) be the largest rectangle with two adjacent vertices on the radius \(3\) circle and two adjacent vertices on the radius \(4\) circle.

Let \(x\) and \(y\) be the dimensions of \(R\), with \(x \gt y\). Then if \(x - y = \dfrac{a}{b}\), where \(a\) and \(b\) are positive coprime integers, find \(a + b\).

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