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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.

User3910
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  • wrong. Circle is obtained if plane is orthogonal to the axis. Ellypsis is obtained if the plane is slightly slanted - less than the envelope line. Parabola is obtained if the plane is slanted in the same angle as that envelope line. And hyperbola is obtained if the plane is heavily slanted - more than the envelope line. - I.e. your assertion of the first sentence, that the plane would be parallel to the axis, is just some example, not the general case! – Dr. Richard Klitzing Mar 27 '18 at 22:15
  • https://web.archive.org/web/20070830171841/http://mathdl.maa.org:80/convergence/1/?pa=content&sa=viewDocument&nodeId=196&bodyId=204 – amd Mar 27 '18 at 22:50

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