a hyperbola
/ hy-PUR-buh-luh /
Hold a lamp near a wall and the cone of light casts a curved edge that flares open without ever closing; the path of a comet swinging past the Sun once and never returning; the graph of y = 1/x. Each is a hyperbola — a conic that comes in two separate, mirror-image branches, each opening outward forever.
Its focal definition mirrors the ellipse's but swaps the sum for a difference: a hyperbola is the set of all points P for which the difference of the two focal distances is constant, | |PF_1| - |PF_2| | = 2a. With foci at (-c, 0) and (c, 0) on the x-axis, the standard equation is x^2/a^2 - y^2/b^2 = 1 (note the minus sign — that is what splits it into two branches), where now c^2 = a^2 + b^2, so c > a. The center is the midpoint of the foci, the two vertices sit at (-a, 0) and (a, 0), and far from the center the branches hug two straight lines called the asymptotes, y = +-(b/a)x, which the curve approaches but never touches.
Why it matters: a hyperbola has eccentricity e > 1, the open end of the conic family. Comets and spacecraft on escape trajectories trace hyperbolas; navigation systems like the old LORAN and GPS pin your position by intersecting hyperbolas of constant time-difference between signals. The hyperbola's reflective property — a ray aimed at the far (virtual) focus reflects toward the near focus — is used in Cassegrain telescopes.
For x^2/16 - y^2/9 = 1, read a^2 = 16, b^2 = 9, so a = 4, b = 3. Then c^2 = a^2 + b^2 = 16 + 9 = 25 gives c = 5: foci at (-5, 0) and (5, 0). The vertices are (-4, 0) and (4, 0), and the asymptotes are y = +-(3/4)x. At the vertex (4, 0), the focal distances are |4 - (-5)| = 9 and |4 - 5| = 1; their difference is 8 = 2a.
Minus sign, two branches, and c^2 = a^2 + b^2 — the hyperbola's signature.
Do not confuse the hyperbola's relation c^2 = a^2 + b^2 with the ellipse's c^2 = a^2 - b^2. For the hyperbola c is the largest (the foci lie beyond the vertices); for the ellipse a is the largest (the foci lie inside).