Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 31B / c / Solution

For $k=1$, the interval in part (b) contains one exactly when $a>1$. With $V=x^2+y^2=r^2$,because $2xy\le r^2$. Hence $\dot{V}<0$ throughoutEvery trajectory starting in the disc $x^2+y^2<a-1$ remains there and approaches the origin. Thus this disc is contained in the basin of attraction of the origin.