= Solution
The quadratic form has symmetric matrix
$$
A=\begin{pmatrix}7&2&1\\2&3&0\\1&0&3\end{pmatrix},
$$
whose characteristic polynomial is $(\lambda-2)(\lambda-3)(\lambda-8)$. All its eigenvalues are positive, so $A$ is <positive-definite matrix>[positive definite] and $\mathbf x^TA\mathbf x=1$ is an <ellipsoid>. The <spectral theorem for real symmetric matrices> gives semi-axis lengths
$$
\boxed{\frac1{\sqrt2},\quad\frac1{\sqrt3},\quad\frac1{\sqrt8}.}
$$
The nearest points lie in the eigendirection of the largest eigenvalue. Since $(5,2,1)^T$ is an $8$-eigenvector, they are
$$
\boxed{\mathbf x=\pm\frac{(5,2,1)^T}{4\sqrt{15}}.}
$$
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