N2009 P1 Q6

The curve \(C_1\) has equation \(y=\frac{x-2}{x+2}\). The curve \(C_2\) has equation \(\frac{x^2}{6}+\frac{y^2}{3}=1\).

1. Sketch \(C_1\) and \(C_2\) on the same diagram, stating the exact coordinates of any points of intersection with the axes and the equations of any asymptotes.[4]
2. Show algebraically that the \(x\)-coordinates of the points of intersection of \(C_1\) and \(C_2\) satisfy the equation \(2(x-2)^2=(x+2)^2(6-x^2)\).[2]
3. Use your calculator to find these \(x\)-coordinates.[2]