Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 320 / 1 / a / Solution

Choose intrinsic Cartesian coordinates in which the oblate spheroid isLet $\ell$ be distance along the line of sight and let $y$ be the sky coordinate in the plane containing the line of sight and the symmetry axis. A rotation through the inclination $i$ givesSubstitution into the ellipsoid equation and minimization over $\ell$, equivalently requiring the quadratic in $\ell$ to have zero discriminant on the projected boundary, givesTherefore the projected axis ratio of an oblate spheroid isIt correctly gives $Q=1$ face-on and $Q=q$ edge-on.