Equation of hyperbola from conic section

[asdfghjkllkjhfdsa]

Is it possible to get an equation of a hyperbola from simply knowing the gradient of a cone positioned with its tip at some positive value of z where x and y are 0, assuming the conic section is made through z=0? What I am trying to explain can be seen in the image sent below, however, the conic section is made therough z=-5 or something, but this was just because I could not put the points for the cone above z=0 for some reason. The red axis is the x axis and the green is the y.

Gyazo

gyazo.com

 

[firemath]

Since all hyperbolas are is back-to-back cones, if you know the gradient of the cone (the slope in 3d space) and the area of it's bottom face, then it would enable you to find the equation of any hyperbola related to that particular cone if the cone is restricted to z=0, like you said. But without the length of the cone or the area of it's bottom face, I think that it would be extremely difficult to get a real equation.