My Problem Number Six

Algebra Level 5

Let \( \mathcal R\) be the set of real numbers and \( \mathcal S\) the set of non-real complex numbers.

If \(a\) is a positive real number and \(x, y\) are in \( \mathcal R\) or \( \mathcal S\) such that \(x^{2}+y^{2}=4a^{2}\), \(x+y=3a\), then which of the following is true?

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