= Solution
Choose intrinsic Cartesian coordinates in which the <oblate spheroid> is
$$
X^2+Y^2+\frac{Z^2}{q^2}=1.
$$
Let $\ell$ 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 $i$ gives
$$
Y=y\cos i-\ell\sin i,
\qquad
Z=y\sin i+\ell\cos i.
$$
Substitution into the ellipsoid equation and minimization over $\ell$, equivalently requiring the quadratic in $\ell$ to have zero discriminant on the projected boundary, gives
$$
X^2+\frac{y^2}{\cos^2i+q^2\sin^2i}=1.
$$
Therefore the <projected axis ratio of an oblate spheroid> is
$$
\boxed{Q^2=\cos^2i+q^2\sin^2i}.
$$
It correctly gives $Q=1$ face-on and $Q=q$ edge-on.
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