Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2023 / Paper 1 / 6 / b / Solution

The quadratic form has real symmetric matrixIts unit eigenvectors can be chosen as $e_u=(1,1,0)/\sqrt{2}$ with eigenvalue $1$, and $e_v=(1,-1,0)/\sqrt{2}$, $e_w=(0,0,1)$ with eigenvalue $9$. SetThe orthogonal diagonalization turns the surface equation intoIt is an ellipsoid, specifically a prolate spheroid centered at the origin. Its long principal axis is the line in direction $\boxed{(1,1,0)/\sqrt{2}}$, with semiaxis length $\boxed{3}$. Each direction in the plane perpendicular to that line has semiaxis length $\boxed{1}$; the displayed $e_v,e_w$ give one orthonormal choice of the two short principal axes. Their nonuniqueness reflects the repeated eigenvalue $9$ and rotational symmetry about the long axis.