2020 TMJC Promo Q4

A curve \(C\) has equation \(y=\frac{2}{{{x}^{2}}-{{k}^{2}}}\) where \(k\ge 2\). It is given that \(C\) has only one stationary point.

1. Find \(\frac{\mathrm{d}y}{\mathrm{d}x}\). Hence find the coordinates of the stationary point in terms of \(k\).[4]
2. Sketch the graph of \(C\), stating the equations of any asymptotes. coordinates of any points of intersection with the axes and the coordinates of the stationary point.[3]
3. By adding a suitable graph on the same diagram in part (ii), find the range of values of \(h\), in terms of \(k\), where \(h\) is a positive real constant and \(h\ne k\) such that \({{\left( hx \right)}^{2}}+{{\left( \frac{2k}{{{x}^{2}}-{{k}^{2}}} \right)}^{2}}={{\left( kh \right)}^{2}}\) has at least one real root.[3]