Centre, axes of 9x²+4xy+6y²−22x−16y+9=0

Here a = 9, h = 2, b = 6, so h² − ab = 4 − 54 < 0: an ellipse.

Centre

9x + 2y − 11 = 0 and 2x + 6y − 8 = 0. Solving: x = 1, y = 1. Centre: (1, 1).

Shift the origin

Constant: g·x₀ + f·y₀ + c = (−11)(1) + (−8)(1) + 9 = −10, so

9X² + 4XY + 6Y² = 10

Remove XY by rotation

The new coefficients are the roots of λ² − 15λ + (54 − 4) = 0, i.e. λ² − 15λ + 50 = 0, so λ = 5 and 10. Hence

5u² + 10v² = 10, or u²/2 + v²/1 = 1

Results

• Semi-axes: √2 and 1. Lengths of axes: 2√2 (major) and 2 (minor).
• Eccentricity: e² = 1 − b²/a² = 1 − 1/2 = 1/2, so e = 1/√2.
• The major axis is along the eigen-direction of λ = 5, i.e. (−1, 2): the line 2x + y − 3 = 0; the minor axis is x − 2y + 1 = 0.