This problem combines several key concepts from coordinate geometry, specifically related to ellipses. We will use the condition for a line to be tangent to an ellipse, find the coordinates of the points of tangency and intersection, and finally calculate the area of the resulting quadrilateral.

1. Key Concept: Condition for Tangency to an Ellipse

For an ellipse given by the standard equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, a line $y = mx + c$ is tangent to the ellipse if and only if the condition $c^2 = a^2m^2 + b^2$ is satisfied. This is a fundamental result for ellipses and is crucial for finding the tangent lines.

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