2018 AJC Promo Q4

The curve \({{C}_{1}}\) has equation \(y=\frac{{{x}^{2}}-5x+10}{x-2}\), \(x\ne 2\). The curve \({{C}_{2}}\) has equation \({{x}^{2}}-{{y}^{2}}=4\).

1. Sketch \({{C}_{1}}\) and \({{C}_{2}}\) on the same diagram, indicating any points of intersection with the axes and the coordinates of any stationary points. Equations of asymptotes should be clearly labelled.[5]
2. Show algebraically that the \(x\)-coordinate of the point of intersection of \({{C}_{1}}\) and \({{C}_{2}}\) satisfies the equation \({{\left( x-2 \right)}^{2}}\left( {{x}^{2}}-4 \right)={{\left( {{x}^{2}}-5x+10 \right)}^{2}}\).[1]
3. Use your calculator to find this \(x\)-coordinate.[1]