2007 SAJC P1 Q7 [Modified]

The curve \(C\) has equation \(y=\frac{a{{x}^{2}}+2}{x-1}\) where \(x\ne 1\) and \(a\) is a non-zero constant.

1. Show that if the curve \(C\) has no stationary points, then \(-2 < a < 0\).
2. Sketch the curve for \(a=1\), showing clearly the asymptotes and coordinates of any intersections with the coordinate axes.
3. Verify that \(y=k(x-1)+2\) passes through \((1,2)\) for all real values of \(k\).
4. By considering the equation of an appropriate line drawn on the same diagram with the curve \(C\), find the range of values of \(k\) for which the equation \({{x}^{2}}+2=k{{(x-1)}^{2}}+2(x-1)\) has no real roots.