1999 TYS

The curve $C$ has equation $y=\frac{{{x}^{2}}+5}{\lambda x+1}$, where $\lambda $ is a non-zero constant.

1. Obtain the equations of the asymptotes of $C$.
2. Show that $C$ has exactly two points at which $\frac{\text{d}y}{\text{d}x}=0$.
3. Draw a sketch of $C$ to illustrate the case $\lambda >0$, and a separate of $C$ to illustrate the case $\lambda <0$.
4. Find the set of values of $\lambda $ for which the line $y=4x$ and $C$ has at least one point in common.