Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / 6 / b

The quadratic-form matrix has eigenvectorswith eigenvalues $1,9,9$. In the principal axes coordinates $\xi =(x_1+x_2)/\sqrt{2}$, $\eta =(x_1-x_2)/\sqrt{2}$ and $\zeta =x_3$, the equation becomesIt is a prolate spheroid, centred at the origin, with semiaxis three along $e_{+}$ and one in every perpendicular direction. The displayed $e_{-},e_3$ are one choice of short principal axes.

The sphere has equation ${\xi }^2+{\eta }^2+{\zeta }^2=4$. Combining it with the ellipsoid equation givesThe intersection of a prolate spheroid and a concentric sphere therefore consists of two circles in planes perpendicular to $e_{+}$. Each has radius $\boxed{\sqrt{5/8}=\sqrt{10}/4}$, and their centres areTheir plane equations in the original coordinates are $x_1+x_2=\pm 3\sqrt{3}/2$.