= Solution
The <characteristic polynomial> is $(5-\lambda)(3-\lambda)-3=(\lambda-6)(\lambda-2)$. Corresponding normalized <eigenvectors> are
$$
\lambda=6:\quad e_6=\frac12(\sqrt3,1),\qquad
\lambda=2:\quad e_2=\frac12(-1,\sqrt3).
$$
These form an <orthonormal basis>. Write $x=u e_6+v e_2$; the <quadratic form> becomes
$$
6u^2+2v^2=4,\qquad\boxed{\frac{u^2}{2/3}+\frac{v^2}{2}=1.}
$$
The locus is an <ellipse> centred at the origin. Its minor semiaxis has length $\sqrt{2/3}$ along the direction $30^\circ$ from the positive horizontal axis; its major semiaxis has length $\sqrt2$ along $120^\circ$. The right panel of the figure in part 1 shows these principal directions.
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