2018 EJC Promo Q9

The curve \(C\) has equation \(y=\frac{{{x}^{2}}+ax-4}{x+b}\), where \(a\) and \(b\) are constants, and \(x\ne -b\). It is given that the asymptotes of \(C\) are \(y=x-2\) and \(x=1\).

1. Find the values of \(a\) and \(b\). Hence sketch \(C\), stating the coordinates of the point(s) where the curve crosses the axes, and the equations of the asymptotes.[5]
2. By drawing a suitable curve on the same diagram in part (i), find the number of real distinct roots of the equation \({{x}^{2}}-2x-20+21{{\left( \frac{{{x}^{2}}+ax-4}{x+b} \right)}^{2}}=0\).[3]