When axes are rotated by angle theta (counterclockwise): x = Xcos(theta) - Ysin(theta) y = Xsin(theta) + Ycos(theta)

Inverse (old to new): X = xcos(theta) + ysin(theta) Y = -xsin(theta) + ycos(theta)

Purpose: To eliminate the xy-term in ax^{2+2hxy+by}^2=... The angle needed: tan(2*theta)=2$\frac{h}{a-b}$. If a=b, then theta=pi/4.

After rotation, the new equation has no cross-term, making conic identification straightforward.

Key: Rotation preserves distances, angles, and areas. It only changes the orientation of the axes.