Supporting the \(\sin\)

Geometry Level 5

Refer to the image above (not drawn to scale)

The \(2\) circles above are tangent to the graph \(y=\sin(x)\). The two circles share \(1\) point of tangency \((a, b)\), and the ratio of the area of the smaller circle to the area of the bigger circle is \(\frac{1}{4}\).

Given that there are two possible values for \(b\), (\({b}_{1}\) and \( {b}_{2}\)), and that \({b}_{1}>{b}_{2}\), and that the sum of the area of both the circles can be expressed as \(A\), find \[\left\lfloor 10000\left( A+ \left( {b}_{1} - {b}_{2}\right)\right) \right\rfloor \]

\({\scriptsize\text{ Congratulate yourself after you solve this}}\)


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