Rotation angle to remove xy term

Angle of rotation

Rotate the axes through θ: x = X cosθ − Y sinθ, y = X sinθ + Y cosθ. For ax² + 2hxy + by², the XY coefficient becomes

2h cos2θ − (a − b) sin2θ

and it vanishes when tan 2θ = 2h/(a − b).

Here a = 3, b = 3, 2h = 2, so a − b = 0 and tan 2θ = ∞. Thus 2θ = 90°, θ = 45°.

Transformed equation

With θ = 45°, x = (X − Y)/√2 and y = (X + Y)/√2.

• x² + y² = X² + Y²
• xy = (X² − Y²)/2
• √2 x = X − Y

Then

3x² + 2xy + 3y² = 3(X² + Y²) + (X² − Y²) = 4X² + 2Y²

and the equation 3x² + 2xy + 3y² − √2 x = 0 becomes

4X² + 2Y² − X + Y = 0

The XY term is gone. Completing the squares gives 4(X − 1/8)² + 2(Y + 1/4)² = 1/16 + 1/8 = 3/16: an ellipse with centre (1/8, −1/4) in the new axes.