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Geometry Level 3

\[ \begin{cases} \dfrac{\sin\theta}{x} =\dfrac{\cos\theta}{y} \\ \dfrac{\cos^{4}\theta}{x^{4}}+\dfrac{\sin^{4}\theta}{y^{4}}=\dfrac{97\sin 2\theta}{x^{3}y+y^{3}x} \end{cases} \]

Let \(x\) and \(y\) be positive real numbers and \(\theta\) is an angle such that it is not a multiple of \(\frac{\pi}{2}\). If \(x,y\) and \(\theta\) satisfy the system of equations above, find \(\dfrac{x}{y}+\dfrac{y}{x}\).


Source: 2009 Harvard-MIT Mathematics Tournament
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