Maximum area of a strange triangle

Geometry Level 3

Consider an ellipse x2144+y264=1\dfrac{x^2}{144} + \dfrac{y^2}{64} = 1. A line is drawn tangent to the ellipse at a point PP. A line segment drawn from the origin to a point QQ on this line is perpendicular to this tangent line.

Find the maximum area of POQ\triangle POQ.

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