2023 RVHS P1 Q9

The curve \(C\) has equation \(y=\frac{a{{x}^{2}}+bx-14}{x+c}\). It is given that the equations of the asymptotes of \(C\) are \(x=3\) and \(y=x+5\).

1. Determine the values of \(a\), \(b\) and \(c\).[3]
2. Sketch the graph of \(C\) indicating clearly the equations of the asymptotes, coordinates of axial intercepts and stationary points.[3]
3. On the same axes, sketch the graph of \(\frac{{{\left( x-3 \right)}^{2}}}{{{m}^{2}}}-{{\left( y-8 \right)}^{2}}=1\), where \(m\) is a positive real constant, indicating clearly the equations of the asymptotes.[2]
4. Hence state the range of values for \(m\) for which the equation \(\frac{{{\left( x-3 \right)}^{2}}}{{{m}^{2}}}-{{\left( \frac{{{x}^{2}}-6x+10}{x-3} \right)}^{2}}=1\) has roots.[1]