2.6: Tangent to curves; perpendicular bisector of $AB$

The line $y = 12 - 2x$ is a tangent to two curves. Each curve has an equation of the form $y = k + 6 + kx - x^{2}$, where $k$ is a constant. The line is a tangent to one curve at the point $A$ and the other curve at the point $B$.