2012 AJC P1 Q9

The curve \(C\) has equation \(y=\frac{2{{x}^{2}}-a}{x+k}\) where \(a,\ \,k>0\).

1. Given that \(2{{k}^{2}}\ne a\), find the range of values of \(k\) such that the curve \(C\) has at least one tangent parallel to the \(x\)-axis.[4]
2. What can be said about the graph of \(y=\frac{2{{x}^{2}}-a}{x+k}\) when \(2{{k}^{2}}=a\)?[2]

For the range of values of \(k\) in (i),

3. Sketch the curve of \(C\), clearly indicating the equations of the asymptotes and any intercepts with the axis.[3]
4. The curve \(y=\frac{2{{x}^{2}}-a}{x+k}\) does not intersect the curve \({{(x+k)}^{2}}-{{\left( \frac{y+4k}{b} \right)}^{2}}=1\) where \(b>0\). By drawing the \(2\) curves on the same diagram, find the range of values of \(b\).[2]