2020 EJC P2 Q2

The curve \({{C}_{1}}\) has equation \(y=\frac{3{{x}^{2}}-2x+3}{x-1}\). The curve \({{C}_{2}}\) has equation \({{\left( x-1 \right)}^{2}}-\frac{{{\left( y-4 \right)}^{2}}}{{{b}^{2}}}=1\), where \(b>0\).

1. Sketch \({{C}_{1}}\), stating the equations of any asymptotes and the coordinates of any turning points and points where the curve crosses the axes.[4]
2. Hence, find the range of values of \(b\) such that there is no point of intersection between \({{C}_{1}}\) and \({{C}_{2}}\). Using the maximum value of \(b\) found, sketch \({{C}_{2}}\) on the same diagram as part (i).[3]