Product of semi-axes of x²−4xy+5y²=2

Here a = 1, h = −2, b = 5, and the centre is the origin (no linear terms), so

x² − 4xy + 5y² = 2

h² − ab = 4 − 5 = −1 < 0, so the conic is an ellipse.

Rotate to remove the xy term

After rotating through the principal angle (tan 2θ = 2h/(a − b) = 1), the equation becomes λ₁u² + λ₂v² = 2, where λ₁, λ₂ are the roots of

λ² − (a + b)λ + (ab − h²) = 0 λ² − 6λ + 1 = 0

so λ₁ + λ₂ = 6 and λ₁λ₂ = 1. (The roots are 3 ± 2√2.)

Semi-axes

u²/(2/λ₁) + v²/(2/λ₂) = 1

Semi-axes: A = √(2/λ₁), B = √(2/λ₂).

Product of semi-axes = AB = √(4/(λ₁λ₂)) = 2/√1 = 2

Individually, A = √(2/(3 − 2√2)) = √2(√2 + 1) = 2 + √2 and B = √2(√2 − 1) = 2 − √2, whose product is 4 − 2 = 2. This agrees.