Rotate axes 45° in 3x²+3y²+2xy=2

For a rotation of the axes through θ, replace

x = X cosθ − Y sinθ, y = X sinθ + Y cosθ

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

Substitution

• x² = (X − Y)²/2, y² = (X + Y)²/2, so x² + y² = X² + Y².
• xy = (X² − Y²)/2.

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

So the equation 3x² + 3y² + 2xy = 2 becomes

4X² + 2Y² = 2, that is 2X² + Y² = 1

(or X²/(1/2) + Y²/1 = 1). The xy term has disappeared; the curve is an ellipse with semi-axes 1/√2 along the new X-axis and 1 along the new Y-axis.