1 · Observe

Drag e slowly from 0 to 2 and watch the shape morph.

2 · Manipulate

3 · Predict

Start at e = 0.5 (an ellipse). Before dragging: what do you think happens exactly at e = 1? And just past it?

4 · See

5 · Explain

Every conic is the set of points where (distance to focus) / (distance to directrix) = e. The polar form r = ed/(1 + e·cosθ) puts the focus at the origin. When e < 1 the denominator never hits zero and the curve closes (ellipse, or a circle when e = 0). At e = 1 the curve just escapes to infinity (parabola); past 1 it splits into two branches (hyperbola).