Equate the curve and line: \[(2k-3)x^2-(k+3)x-(k-6)=0.\] For tangency, the quadratic must have a repeated root. Thus \[(k+3)^2+4(2k-3)(k-6)=0\] \[9k^2-54k+81=9(k-3)^2=0.\] Therefore \[\boxed{k=3}.\] With this value the difference of curve and line is \(3(x-1)^2\), confirming tangency at \((1,-1)\). The exceptional value \(k=3/2\) gives a distinct straight line and cannot satisfy the tangency condition.