rotation of axes
When a conic's equation contains an xy term, the curve is tilted — its axis of symmetry runs at a slant to the coordinate axes, and the equation is awkward to read. Rotation of axes is the trick of spinning the coordinate system to line up with the tilted conic, after which the cross term vanishes and the equation snaps back into a clean, recognizable standard form.
Here is the method. Replace the old coordinates (x, y) by new ones (x', y') turned through an angle theta: x = x' cos theta - y' sin theta and y = x' sin theta + y' cos theta. The angle is chosen to kill the cross term, which happens when cot(2 theta) = (A - C)/B (equivalently tan(2 theta) = B/(A - C)). Plug the substitution into Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, collect terms, and the new B' coefficient becomes zero by that exact choice of theta. The equation is now A'x'^2 + C'y'^2 + (lower terms) = 0 in the rotated frame, which you finish by completing the square as usual. The conic itself never moved — only your viewpoint rotated to meet it.
Rotation of axes is what completes the analysis of the general second-degree equation: the discriminant tells you the type, and rotation (plus translation) puts any conic, however tilted and shifted, into a standard equation you can read foci and axes from. The quantities A + C and B^2 - 4AC stay fixed through the rotation, which is both a useful check and the reason the discriminant classification is trustworthy.
Take xy = 1, so A = 0, B = 1, C = 0. Here A = C, so 2 theta = 90 degrees and theta = 45 degrees. Substituting x = (x' - y')/sqrt(2), y = (x' + y')/sqrt(2) turns xy = 1 into (x'^2 - y'^2)/2 = 1, that is x'^2/2 - y'^2/2 = 1 — plainly a rectangular hyperbola, now sitting squarely on the new axes.
Spin the axes by theta with tan(2 theta) = B/(A - C) and the xy term disappears.
Rotation of axes moves your coordinate frame, not the curve — the conic is identical before and after, only described by simpler numbers. When A = C the cross term equation gives theta = 45 degrees directly, since tan(2 theta) blows up.