Cho \({\log _9}x = {\log _{12}}y = {\log _{16}}\left( {x + 3y} \right).\) Tính giá trị \(\frac{x}{y}\)

A. \(\frac{{3 - \sqrt 5 }}{2}\)

B. \(\frac{{\sqrt 5 - 1}}{2}\)

C. \(\frac{{3 + \sqrt {13} }}{2}\)

D. \(\frac{{\sqrt {13} - 3}}{2}\)