I am able to follow the procedure of rewriting a general conic section in standard form but have a question concerning what is actually happening geometrically at each step. I will break these down into several parts:
1, The general equation of the conic section is $ax^2 + bxy + cy^2 + dx + ey + f =0$. How can we understand this as being a slice of a cone by a plane? When I google the picture of the various conic sections available, we have this double cone whose equation is presumably $X^2+Y^2=|Z|$ where I've introduced $(X,Y,Z)$ as coordinates on an ambient 3d space whilst $(x,y)$ are coordinates on the 2d plane. The equation of the plane in the ambient 3d space is going to be like $AX+BY+CZ=Q$. Now, if I substitute $Z=\frac{1}{C(Q-AX-BY)}$ into the cone equation and relabel the constants, I can get something like $X^2+Y^2+DX+EY+F=0$ but this has no $XY$ cross term, nor do I understand how to get it in terms of the $(x,y)$ coordinates that exist on the 2d planar surface (and thus that actually describe the conic section).
TLDR: Where does the general equation of the conic section come from? Can somebody convince me that it is indeed the intersection of a cone and a plane?
2, We can then rewrite $ax^2+bxy+cy^2+dx+ey+f=0$ as $\vec{x}^T A \vec{x} + \vec{v}^T \cdot \vec{x} + f=0$. My question here is what does each part describe? Intuitively I thought that $\vec{v}^T \cdot \vec{x}$ represents a translation of the cone within the ambient 3d space such that we get the same conic section but "off=centre" in terms of the 2d coordinates $(x,y)$. Is this true? If I have $b=0$ and keep the plane in the same place but move the cone then surely their intersection will not only have a different centre, but it will also have a different size i.e. it will be the same shape of section but either scaled up or down - do people agree with this? Then surely this term needs to describe more than just a translation but also a scaling - does this happen?
3, What about the case where $b \neq 0$. In this case, I have read that this represents a rotation of the conic section. What causes this - does the cone rotate or the plane? This is a part that I really struggle to get my head around since the matrix $A$ does not have unit determinant and therefore doesn't represent a rotation matrix, does it? Typically we would construct the matrix of orthonormal eigenvectors $P$ which would have unit determinant and use this to describe a rotation to a new coordinate system $(x',y')$ in which the conic section assumes the standard form. As I understand it, the symmetry of $A$ means that there will always exist such a $P$ that orthogonally diagonalises $A$ (spectral theorem) and thus $P$ describes a rotation from $(x,y)$ to $(x',y')$ where $x'$ and $y'$ point in the eigendirections of $A$ i.e. the eigendirections are the new major and minor axes for the conic section when written in $(x',y')$ coordinates. BUT if that's true, then P describes a rotation and A describes WHAT EXACTLY????
4, How can I understand why the eigendirections of $A$ will be the major and minor axes of the conic section in the $(x',y')$ coordinates?
THank you very much for your help. I am just struggling to visualise what's actually happening behind the relatively straightforward mathematics here.