Rotating a Hyperbola: The Principal Axes Problem

We provide an example of the principal axes problem for one of the most elementary functions: the reciprocal function. Why is the graph of y=1/(2x) a hyperbola? We write this function in the implicit form of 2xy=1, then use quadratic forms to orthogonally diagonalize the principal part. The similarity matrix is orthogonal, so in the rotated coordinates the quadratic form has the standard form of a hyperbola!
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