Rotation of axes
Theorem 6.1 (Rotation of axes). Rotating the axes through an angle \(\theta \) gives new coordinates \((x', y')\) related to the old ones by
\[ \begin {aligned} x &= x' \cos \theta - y' \sin \theta , & x' &= x \cos \theta + y \sin \theta , \\ y &= x' \sin \theta + y' \cos \theta , & y' &= -x \sin \theta + y \cos \theta . \end {aligned} \]
If \(B \neq 0\), the rotation with
\[ \cot 2\theta = \frac {A - C}{B}, \qquad 0 < \theta < \frac {\pi }{2}, \]
removes the cross term from \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\). In particular \(\theta = \pi /4\) when \(A = C\).
Proof. The two systems are inverse to each other, since the matrix of a rotation by \(\theta \) has inverse the rotation by \(-\theta \). Substitute the left-hand formulas into the quadratic part. Only \(x^2\), \(xy\) and \(y^2\) produce an \(x'y'\) term, and its coefficient is
\[ -2A \sin \theta \cos \theta + B(\cos ^2\theta - \sin ^2\theta ) + 2C \sin \theta \cos \theta = B\cos 2\theta - (A - C)\sin 2\theta . \]
This vanishes exactly when \(\cot 2\theta = (A - C)/B\), and \(\cot \) takes every real value once on \((0, \pi )\).