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 \(a + b + c\).