Rotation of Axes

Algebra 2 · Axiom Academy

A cross term tilts a conic — rotate the axes to the right angle and it snaps back to standard form. An equation with no cross term ( B = 0 ) lines up with the axes. Switch on an xy term and the same conic rotates off them. These two equations describe the same ellipse , once lined up with the axes and once rotated . 2. Rotate the Axes to Remove It The conic is fine — our axes are the problem. Rotate them by an angle so the new x',y' axes line up with the conic's own axes of symmetry, and the cross term drops to zero. The angle that does it, and the substitution that carries the equation into the new frame: Pick from the coefficients A , B , C Substitute to rewrite the conic in the x',y' frame Take the rectangular hyperbola xy = 1 . Here A = 0 , C = 0 , and B = 1 , so , giving and . Watch that rotation carry it into standard position: A standard east–west hyperbola, vertices at — and no xy term left. B^2 - 4AC < 0 — ellipse (or circle) You've seen why an xy term tilts a conic, and how one rotation of the axes clears it away. Scroll up to revisit any step.

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