Tilted Conics

Rotating away the xy term

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• An xy term means the conic is tilted.
• Rotate by θ with tan 2θ = B/(A − C) to remove it.
• B² − 4AC < 0 ellipse, = 0 parabola, > 0 hyperbola, unchanged by rotation.

Steps

1. The general equation
2. What is xy = 1?
3. Line up the axes
4. The discriminant
5. Classify it
6. The longest reach