All of us know general equation for any conic section:
$$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$
If we'll change of parameters of this equation we get different type of conic section. If we imagine behavior of plane as section plane, there will be not intuitievly.
For example we get this:
If we change C parameteres from 1 to 2 we get this:
I try to figure out how a changing of a posititon of section plane will change of section line from 1 to 2,a and I can't.
Is there the equation of section plane in 3D which describes of behaviour of the plane based on their parameters, where we able to change parameters of equation and see when plane parallel x or y or conic side, to see the section line on conic?
Update:
Thank you Chrystomath for your example. I am really surprised that it takes not two cones to get cross-sections from pictures above, but some other shape (hyperbolic paraboloid). That figure is a part of two cones? It seemed to me that changing parameters A, B, C, etc. we can to control the section plane (blue), and that there is a mathematical model where its behavior in 3D would be related to the first equation clearly, that a changing the parameters (A,B,C) could be visually show the position of the section plane in space and the section lines. Below is roughly what I need.
Ellipse/circle.
Hyperbola
Update 2:
Here is the rough equation I able to figure out for any conic section!
One can take the parameter itself to be the $z$ variable and thus the conics will by the $xy$-cross sections for constant $z$.
For the equations used in the question, consider $$xy+x+y+z=0$$ with $z$ varying between $0$ and $2$. The resulting surface is called a hyperbolic paraboloid. It is drawn below together with the cross-sections $z=0$ (blue) and $z=2$ (red). The curve for $z=1$ (black) is the cross-over from one curve to the other. A view from above is also given.
[Note: The surface and curves are rotated versions of $xy+x+y+z=0$, in order to make more of the surface visible.]