Hyperbolas are made whenever a plane is normal to the radius of rotation. Which hyperbola is formed is dependent on the radius and a scaling factor. What radius R and scaling factor would be needed for 1/x? assuming: https://www.quora.com/What-is-the-equation-of-a-circular-cone
$$x^2+y^2=a^2⋅z^2$$ suppose R is in the x direction, then x=R $$R^2+y^2=a^2⋅z^2$$ $$R^2=a^2⋅z^2-y^2$$ $$R=(a^2⋅z^2-y^2)^{1/2}$$
Or $r=a*z$ and $r=b$ then it is slightly more complicated.