Answer to Question #114939 in Analytic Geometry for ANJU JAYACHANDRAN

Question

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

Answer 1

$x^2+2z^2=y$

Let's rewrite equation: $2x^2+4z^2=2y\iff \frac{x^2}{0.5}+\frac{z^2}{0.25}=2y$

Now we can see that our surface is elliptic paraboloid.

Denominators are unequal. It means that our paraboloid is not a paraboloid of revolution.

Let $x=0$, then $y=2z^2$

And let $z=0$, then $y=x^2$

$y=2z^2$ and $y=x^2$ principal parabolas.

Let's look on sections.

$y=c$, where $c\isin R$

$x^2+2y^2=c -$ ellipses if $c>0$

If $x=c,$ where $c\in R$

$y=2z^2+c^2 -$ parabolas.

$c^2 -$ is distance between the origin and vertex of parabola.

A similar result will be if we look at $z=c,$ where $c\in R$

$y=x^2+2c^2$