= Solution
The <quadratic form> has real <symmetric matrix>
$$
A=\begin{pmatrix}5&-4&0\\-4&5&0\\0&0&9\end{pmatrix}.
$$
Its unit <eigenvectors> can be chosen as $e_u=(1,1,0)/\sqrt2$ with <eigenvalue> $1$, and $e_v=(1,-1,0)/\sqrt2$, $e_w=(0,0,1)$ with <eigenvalue> $9$. Set
$$
u=\frac{x_1+x_2}{\sqrt2},\qquad v=\frac{x_1-x_2}{\sqrt2},\qquad w=x_3.
$$
The <orthogonal diagonalization> turns the surface equation into
$$
u^2+9v^2+9w^2=9,\qquad\boxed{\frac{u^2}{9}+v^2+w^2=1.}
$$
It is an <ellipsoid>, specifically a <prolate spheroid> centered at the origin. Its long principal axis is the line in direction $\boxed{(1,1,0)/\sqrt2}$, with semiaxis length $\boxed{3}$. Each direction in the plane perpendicular to that line has semiaxis length $\boxed{1}$; the displayed $e_v,e_w$ give one orthonormal choice of the two short principal axes. Their nonuniqueness reflects the repeated <eigenvalue> $9$ and rotational symmetry about the long axis.
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