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

Geometry Level 2

Given the ellipse 3x212x+2y2+12y+6=03x^{2} - 12x + 2y^{2} + 12y + 6 = 0, there exists a real number k=ab+ck = a\sqrt{b} + c, (where a,b,ca,b,c are all positive integers and bb is square-free), such that the hyperbola xy2y+3x=kxy - 2y + 3x = k is tangent to the ellipse at two points.

Find a+b+ca + b + c.

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