What is the surface $25x^2 + 75y^2 + z^2 = 0$?

Question

Find the intersection curves of the surfaces $$25x^2 + 25y^2 - z^2 = 25$$ and $$25x^2 + 75y^2 +z^2 = 0\,.$$

I am fairly certain the first surface is a hyperboloid of one sheet, but I cannot figure out what type of surface the second equation represents. It seems to me that it's just a point, but within the context of the question that seems wrong.

Answer 1

$x^2\ge0~\forall~ x\in\mathbb R\Rightarrow 25x^2+75y^2+z^2\ge0\ \forall~ x,y,z\in\mathbb R$. Since $25x^2+75y^2+z^2=0$, it means $(x,y,z)\equiv(0,0,0)$ which the hyperboloid doesn't have on it.

Either there is a typo in your question(probably it is $z$, not $z^2$), or the intersection is in the complex plane.

Answer 2

It is a point, if we are assuming that the domain of the polynomial function $25 x^2 + 75 y^2 + z^2$ is $\Bbb R^3$. However, this can change if $x,y,z$ are assuming values in some other ring or field.

Answer 3

No real intersections because sum of three positive terms is zero, is a null ellipsoid. Instead of $0,$ the RHS may be >0, like 50,75 etc. The post may close if you do not make a change.