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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.

Yes. Even under random orientation, a population concentrated at one nonzero intrinsic ratio $q_0$ haswhich decreases from an integrable divergence at the lowest allowed value $Q=q_0$. It is therefore skewed toward the lower end of its support. More generally, a sufficiently narrow intrinsic distribution can retain such low-$Q$ peaks after the mixture in part d.

If the random-orientation assumption is relaxed, preferentially edge-on selection makes $\cos i$ concentrate near zero and hence makes $Q$ concentrate near $q$. Dust extinction, surface-brightness selection, or alignment by environment can instead bias the sample in either direction. Triaxial or prolate galaxy shapes also invalidate the projected axis ratio of an oblate spheroid formula and can produce other low-$Q$ distributions.