Question

Trace the conicoid represented by x
2 +2z
2 = y. Also describe its sections by the
planes x = c,∀c ∈ R.

Answer 1

$x^2+2z^2=y$  is a paraboloid with vertex at (0, 0, 0) opening away from the origin centered on the line $x=0, z=0$.

The conicoid cross section with planes perpendicular to its centered axis are ellipses with the formula $\frac {x^2}{2}+z^2=\frac{y}{2}$(fig.1 ) The conicoid is the Elliptical paraboloid (fig.2).

When $y=0$, $x^2+2z^2=0$ , which means , so that this section is a single point, (0, 0, 0).

When $x=0$, $y=2z^2$ which is a parabola in the $YZ$ plane with vertex at (0, 0, 0).

When $z=0, y=x^2$ , and this section is a parabola in the $YX$ plane with vertex at (0, 0, 0).

The sections of a paraboloid by the planes $x = c,∀c ∈ R$ .

are parabolas $y=2z^2+c^2$ which are shifted as a whole in the direction of large values of $y$ if $|c|$ is increased. For $x=c$ and $x=-c$ parabolas, in the corresponding planes are the same.

fig.1

fig.2