the focus-directrix property
Here is a single rule that produces all three conics at once, just by turning one dial. Fix a point F (the focus) and a line d (the directrix). For any point P, compare two distances: how far P is from F, and how far P is from the line d. The focus-directrix property says a conic is the set of all points where the ratio of these two distances is a fixed number e, called the eccentricity.
In symbols: P lies on the conic when |PF| / (distance from P to d) = e, a constant. Now turn the dial e. If e = 1 the two distances are always equal and you get a parabola — this is the parabola's own definition. If 0 < e < 1 the point must stay relatively closer to the focus, and the curve closes into an ellipse. If e > 1 the point stays relatively closer to the directrix, and the curve flares open into a hyperbola. (The circle, e = 0, is the limiting case where the directrix runs off to infinity.) Each branch of a conic gets its own focus-directrix pair; an ellipse and a hyperbola have two of each, one for each focus.
This is the great unifier of the conics: a single focus, a single line, and a single number e do the whole job, with the curve's type decided purely by where e sits relative to 1. It is also the bridge to the polar equation of a conic, where this exact ratio becomes r = e d / (1 +- e cos theta) with the focus at the pole.
Let F = (1, 0), directrix x = 4, and e = 1/2 (so an ellipse). For P = (2, y), |PF| = sqrt((2-1)^2 + y^2) and distance to the directrix is |2 - 4| = 2. The condition |PF| = e * 2 = 1 forces sqrt(1 + y^2) = 1, giving y = 0: the point (2, 0) lies on this ellipse, and it is in fact a vertex.
One focus, one line, one ratio e — the value of e alone names the curve.
The circle is the awkward exception: it has no real directrix at finite distance, because e = 0 would force the directrix infinitely far away. So the focus-directrix property describes parabolas, ellipses, and hyperbolas cleanly, but the circle is best handled as the e = 0 limit.