Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / 6 / ii / b / Solution

Let ${\lambda }_1={\lambda }_2=\lambda >0$ and ${\lambda }_3=-\gamma <0$. ThenThis is a one-sheet hyperboloid of revolution around the $x_3^{\prime }$ axis. Its circular waist has radius $\sqrt{k/\lambda }$ at $x_3^{\prime }=0$, and its radius grows with $|x_3^{\prime }|$. In the original coordinates the axis is the eigenvector direction belonging to the negative eigenvalue.