Solution
= Solution
An <orthogonal transformation> to the <eigenvectors> preserves distance from the centre and writes the surface as $u^2+v^2+3w^2=3$. It is an <ellipsoid> with <semi-axes> $\sqrt3,\sqrt3,1$. Therefore
$$
\boxed{\text{nearest distance from }\mathbf a=1.}
$$
The original $y$ axis is one of the eigen-directions with eigenvalue one, and the centre has $a_y=0$. Consequently $y^2\le3$, attained when the other centred coordinates vanish, so $\boxed{y_{\max}=\sqrt3}$.