One Rule, Many Curves

Unified focus-directrix rule: for every point P on the curve, d(P, focus) / d(P, directrix) = e. Slide e across 0 to 2 to move from circle/ellipse to parabola to hyperbola.

Teaching Notes

Core Idea

The same focus-directrix ratio rule generates all conics when e changes.

Try This

Pause at e=1 and ask students why this boundary case is exactly the parabola.

Teacher Prompt

Have students explain the full story: circle/ellipse (e<1), parabola (e=1), hyperbola (e>1).

Vocabulary

Unified model: one equation describing many curves.
Semi-latus rectum: conic scale parameter.
Boundary case: transition point between regimes.