Common surface between two equations

Question

What is common surface between: $(x+5)^2+z^2=y$ and $z^2+y^2=25$ ?

I have found that at the $XY$ plane the common surface is a hyperbola, but it cannot be right because at the paraboloid there isn't any negative $Y$ volume.

Answer 1

Each of those equations defines a surface in 3D space: the first is a paraboloid and the second is a cylinder. Their intersection is not a "surface" but rather a curve, rather like a circle that is warped to fit the surface of the cylinder. Here is a graph:

Note that the $xy$ plane is rotated to get a better view: positive $x$ is up and to the left, positive $y$ and down and to the left.

You can see that the $xy$ plane is basically irrelevant to the intersection curve, so your analysis there is off-point.

There are multiple ways to define the intersection curve. The best may be a parameterization in one variable. Here is one parameterization in $z$, giving two points for each value of $z$.

$$-\sqrt{\frac{-1+\sqrt{101}}2} \le z \le \sqrt{\frac{-1+\sqrt{101}}2}$$ $$y=\sqrt{25-z^2}$$ $$x=-5\pm \sqrt{y-z^2}$$