= Solution
{c}
The <quadric surface> $x^2+y^2+z^2=1$ is the unit <sphere> centered at the origin. Each section $z=c$, for $|c|<1$, is a <circle> of <radius> $\sqrt{1-c^2}$.
The <quadric surface> $z=x^2+y^2$ is a circular <elliptic paraboloid>, opening in the positive $z$ direction with vertex at the origin. Its horizontal sections $z=c>0$ are <circles> of <radius> $\sqrt c$, and its vertical sections through the $z$ axis are <parabolas>.
For $ax^2+y^2=0$, $z$ is unrestricted. If $a>0$, both nonnegative terms must vanish, so the locus is \b[the $z$ axis], a <straight line>. If $a=0$, it is \b[the <plane> $y=0$], represented by the repeated factor $y^2$. If $a<0$, factorization gives
$$
(y-\sqrt{-a}\,x)(y+\sqrt{-a}\,x)=0,
$$
so the locus is \b[two distinct <planes>, $y=\pm\sqrt{-a}\,x$, meeting along the $z$ axis]. These are degenerate <quadrics>, not ordinary curved cylinders.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-iv-q2-quadrics.png]
{title=The <sphere>, <elliptic paraboloid>, and three degenerate <quadric> cases in Question 2(a)}
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