= Solution
In the <orthonormal basis> of part (b), the three <quadric surfaces> have the following forms. For $\alpha=1$,
$$
\frac{u^2}{1/6}-\frac{v^2}{1/3}-\frac{w^2}{1/4}=1,
$$
a <two-sheeted hyperboloid> with axis $u$, vertices $(\pm1/\sqrt6,0,0)$, and one sheet on each side of the plane $u=0$. For $\alpha=0$,
$$
6u^2=3v^2+4w^2,
$$
an <elliptic cone> with vertex at the origin and the same axis. For $\alpha=-1$,
$$
\frac{v^2}{1/3}+\frac{w^2}{1/4}-\frac{u^2}{1/6}=1,
$$
a <one-sheet hyperboloid> with axis $u$ and a waist <ellipse> at $u=0$, whose semiaxes are $1/\sqrt3$ along $v$ and $1/2$ along $w$. The sections at fixed $u$ enlarge away from the waist.
\Image[/past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-1-quadrics.png]
{title=The <two-sheeted hyperboloid>, <elliptic cone> and <one-sheet hyperboloid> in separate orthonormal coordinate frames}
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