Rotation of Axes: Eliminating the xy Term

When a conic equation has an $xy$ term (like $x^{2} + 4xy + y^{2} = 1$), the conic is tilted relative to the axes. To identify the conic and graph it cleanly, we rotate the coordinate system by an angle $\theta$ that eliminates the $xy$ term.

For the general conic $Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0$, the rotation angle is:

$\theta = \dfrac{1}{2}\arctan\!\left(\dfrac{B}{A - C}\right)$

For $x^{2} + 4xy + y^{2} = 1$: $A = 1$, $B = 4$, $C = 1$. Since $A = C$, the formula gives $\theta = 45°$.

After rotating by $45°$: the equation becomes $3X^{2} - Y^{2} = 1$ in the rotated coordinates — a hyperbola with semi-transverse axis $a = 1/\sqrt{3}$ along the rotated $X$-axis.

The discriminant test

$B^{2} - 4AC = 16 - 4 = 12 > 0$ → hyperbola. This confirms the rotation result without doing the full calculation.

Try it in the visualization

Adjust $A$, $B$, $C$ and watch the conic rotate. The original axes (gray) and rotated axes (colored) are shown simultaneously. The discriminant test identifies the conic type. Toggle "show rotated equation" to see the simplified form without the $xy$ term.