Rotate axes 30° in x²+2√3xy−y²=2a²

For a rotation of the axes through θ = 30°:

x = X cos30° − Y sin30° = (√3 X − Y)/2 y = X sin30° + Y cos30° = (X + √3 Y)/2

Substitute

• x² = (3X² − 2√3 XY + Y²)/4
• y² = (X² + 2√3 XY + 3Y²)/4
• xy = (√3X − Y)(X + √3Y)/4 = (√3X² + 2XY − √3Y²)/4

Then

• x² − y² = (2X² − 4√3 XY − 2Y²)/4
• 2√3 xy = (6X² + 4√3 XY − 6Y²)/4

Adding:

x² + 2√3xy − y² = (8X² − 8Y²)/4 = 2X² − 2Y²

So 2X² − 2Y² = 2a², that is

X² − Y² = a²

The xy term has gone: the curve is a rectangular hyperbola with semi-axes a and a.