= Solution
The equality <gradients> are independent everywhere. At an interior point of the ellipsoid, there is no active inequality, so the <Mangasarian–Fromovitz constraint qualification> holds. At a boundary point $x=x_0+z$, a qualifying direction must satisfy $Av=0$ and $2x^TQv<0$.
If $\rho^2<1$, choose $v=x_0-x=-z$. At a boundary point $z^TQz=1-\rho^2>0$, and
$$
2x^TQv=-2z^TQz<0.
$$
Thus the qualification holds at every feasible point. If $\rho^2=1$, the only feasible point is $x_0$; for every $v\in\ker A$, $x_0^TQv=0$, so no strict direction exists. This proves
$$
\boxed{\text{MFCQ at one feasible point}\iff\rho^2<1
\iff\text{MFCQ at every feasible point}.}
$$
The equivalence also identifies this case with strict feasibility of the inequality on the affine equality space.
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