According to the sources, the ellipsoid formula is: $$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$ And the formula for a 2-sheeted hyperboloid is: $$-\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$ Now I have this problem with its respective solution:
$F(x,y)=\sqrt{1-x^2-y^2}$
This surface is the lower part of a hyperboloid of two sheets; the function's domain is $\mathbb R^2$ and its range is the interval $(-\infty,-1]$.
But if you rearrange the equation you would get $$z^2=1-x^2-y^2$$ $$1=x^2+y^2+z^2$$ Which is the formula for an ellipsoid, not a hyperboloid. But I checked the graph of the answer with software and it's correct. How is this possible?
This seems to be merely a typo. To reproduce the given hyperboloid and domain and range the surface should be $$z=-\sqrt{1+x^2+y^2}$$ A more prosaic reason for the error is that the function does not actually refer to the plot.