Area of a Region

Geometry Level 5

The graph of \({\bf 4 x^2 + 12 xy + 9 y^2 + 8 \sqrt{13} x + 12 \sqrt{13} y - 65 = 0 }\) are two parallel lines.

If \({\bf 4 x^2 + 12 xy + 9 y^2 + 8 \sqrt{13} x + 12 \sqrt{13} y - 65 = 0 }\) and the circle \({\bf x^2 + y^2 = 36 }\) intersect at four points and the area of the enclosed trapezoid formed inside the circle can be represented by \({\bf a * (\sqrt{b} + \sqrt{c} ) }\), where \({\bf a, \: b,\:, and \: c }\) are coprime positive integers.

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Find: \({\bf a + b + c. }\)

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