Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics IV / 2 / a / Solution

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 $a{x}^2+y^2=0$, $z$ is unrestricted. If $a>0$, both nonnegative terms must vanish, so the locus is the $z$ axis, a straight line. If $a=0$, it is the plane $y=0$, represented by the repeated factor $y^2$. If $a<0$, factorization givesso the locus is two distinct planes, $y=\pm \sqrt{-a}x$, meeting along the $z$ axis. These are degenerate quadrics, not ordinary curved cylinders.