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

Geometry Level 2

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 $a + b + c$ .