= Solution
Put $D=\operatorname{diag}(a_1,\ldots,a_{n+1})$. The invertible <linear map> $x\mapsto u=Dx$ takes the <ellipsoid> to the unit <sphere> $S^n$. Write $u=(v,s)$ with $v\in\mathbb R^n$, and let $N=(0,1)$ and $S=(0,-1)$ be its two poles. The following two <stereographic projections> define <manifold charts> on the ellipsoid:
$$
U_N=\mathcal Q_n\setminus\{D^{-1}N\},\qquad \chi_N(x)=\frac{v}{1-s},\qquad U_S=\mathcal Q_n\setminus\{D^{-1}S\},\qquad \chi_S(x)=\frac{v}{1+s}.
$$
Their ranges are $\mathbb R^n$. With $r^2=|w|^2$, their inverses, followed by $D^{-1}$, are
$$
\chi_N^{-1}(w)=D^{-1}\left(\frac{2w}{1+r^2},\frac{r^2-1}{1+r^2}\right),\qquad \chi_S^{-1}(w)=D^{-1}\left(\frac{2w}{1+r^2},\frac{1-r^2}{1+r^2}\right).
$$
These are continuous and <smooth functions>, and directly satisfy the defining constraint. On the overlap, the <smooth transition map> is
$$
\boxed{\chi_S\circ\chi_N^{-1}(w)=\frac{w}{|w|^2}\quad(w\ne0).}
$$
It is its own inverse and is smooth away from zero. The two domains are open in the <subspace topology> and cover the ellipsoid. Its <subspace topology> is inherited from <Euclidean space>, so the ellipsoid is <Hausdorff> and has a <second-countable space> topology. Thus these charts form a <smooth atlas> and make it an $n$-dimensional <smooth manifold>, with an explicit <diffeomorphism> to $S^n$. For the degenerate dimension $n=0$, each chart consists of a single point with range $\mathbb R^0$, and their overlap is empty; the same conclusion holds.
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