If one applies an invertible linear transformation to a sphere, one obtains an ellipsoid; it can be brought into the above standard form by a suitable rotation, a consequence of the spectral theorem.
The intersection of an ellipsoid with a plane is empty, a single point or an ellipse.
One can also define ellipsoids in higher dimensions, as the images of spheres under invertible linear transformations. The spectral theorem can again be used to obtain a standard equation akin to the one given above.
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