Maximum area of a strange triangle

Geometry Level 4

Consider an ellipse \(\dfrac{x^2}{144} + \dfrac{y^2}{64} = 1\). A line is drawn tangent to the ellipse at a point \(P\). A line segment drawn from the origin to a point \(Q\) on this line is perpendicular to this tangent line.

Find the maximum area of \(\triangle POQ\).

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