2012 NYJC Promo Q11

The curve \(C\) has equation \(y=\frac{{{x}^{2}}+kx+1}{x-2}\), where \(k\) is a constant.

1. Given that \(y=x\) is an asymptote of \(C\), find the value of \(k\).[2]
2. Let \(k=-2\). Using an algebraic method, find the set of values of \(y\) for which there are no points on \(C\).[3]
3. Sketch the graph of \(y=\frac{{{x}^{2}}+kx+1}{x-2}\) for \(k=-2\), stating clearly the equations of any asymptotes, coordinates of any turning points and axial intercepts.[3]
4. On the same diagram, sketch the graph of \(9{{y}^{2}}=9{{a}^{2}}-{{a}^{2}}{{\left( x-3 \right)}^{2}}\), where \(a\) is a positive constant, showing clearly the coordinates of the axial intercepts.[3]
5. Deduce the value of \(a\) if the equation \(9{{\left( x-1 \right)}^{4}}={{a}^{2}}{{\left( x-2 \right)}^{2}}\left( 9-{{\left( x-3 \right)}^{2}} \right)\) has exactly \(3\) real roots.[2]