A Funny Thing Happened On The Way To Infinity...

Geometry Level 3

Given the ellipse $$3x^{2} - 12x + 2y^{2} + 12y + 6 = 0$$, there exists a real number $$k = a\sqrt{b} + c$$, (where $$a,b,c$$ are all positive integers and $$b$$ is square-free), such that the hyperbola $$xy - 2y + 3x = k$$ is tangent to the ellipse at two points.

Find $$a + b + c$$.

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