Hyperbola Calculator — Vertices, Foci & Asymptotes

Hyperbola Guide

What do I need to know about Hyperbola Properties?

Horizontal hyperbola x²/a² − y²/b² = 1: opens left and right. Vertices: (±a, 0). Foci: (±c, 0) where c = √(a² + b²). Asymptotes: y = ±(b/a)x. Vertical hyperbola y²/a² − x²/b² = 1: opens up and down. Vertices: (0, ±a). Foci: (0, ±c). Asymptotes: y = ±(a/b)x. Key identity (different from ellipse): c² = a² + b² (ellipse: c² = a² − b²). For hyperbola, c > a always. Eccentricity: e = c/a > 1 always (vs ellipse: 0 < e < 1, circle: e = 0).

What's the key thing to understand about Eccentricity?

Eccentricity e = c/a measures how stretched a conic section is. For a hyperbola: e > 1 always. Close to 1: narrow hyperbola (almost like two lines). Large e: very wide, flat hyperbola. Rectangular hyperbola (e = √2): asymptotes are perpendicular (y = ±x for a = b). xy = k is also a rectangular hyperbola (rotated 45°). Applications: Rutherford scattering — alpha particles fired at a gold foil follow hyperbolic paths due to Coulomb repulsion. LORAN navigation used hyperbolic position lines. Conic sections — circles, ellipses, parabolas, and hyperbolas — all share this same eccentricity-based classification, differing only in how elongated or open their curve is.

What should I know about Asymptotes?

Asymptotes are the lines the hyperbola approaches but never touches. For x²/a² − y²/b² = 1: asymptotes y = ±(b/a)x. For y²/a² − x²/b² = 1: asymptotes y = ±(a/b)x. The rectangle formed by the vertices and co-vertices (b, 0) has the asymptotes as diagonals — this is the easiest way to draw asymptotes by hand. A hyperbola with shifted centre (h, k): asymptotes become y − k = ±(b/a)(x − h). The asymptotes and the latus rectum (the chord through a focus perpendicular to the transverse axis) are key features to identify and sketch correctly when graphing a hyperbola by hand.

What do I need to know about Relationship to Other Conics?

All conic sections can be defined as the set of points where the ratio of distance to focus to distance to directrix = eccentricity (e). Circle: e = 0 (no directrix — centre is equidistant from all points). Ellipse: 0 < e < 1 (directrix outside the curve). Parabola: e = 1 (directrix is perpendicular to axis, parallel to the opening). Hyperbola: e > 1 (directrix between the vertices). All four conics arise from intersecting a cone with a plane at different angles. This unified definition, attributed to the ancient Greek mathematician Apollonius of Perga, elegantly connects all four conic sections through a single geometric idea.

What should I know about the Hyperbola's Standard Equation and Vertices?

A horizontal hyperbola centred at the origin follows x²/a² − y²/b² = 1, with vertices at (±a, 0) — the closest points on the curve to the centre. A vertical hyperbola follows y²/a² − x²/b² = 1, with vertices at (0, ±a). The key difference from an ellipse's equation is the minus sign, which is what makes the curve open outward into two separate branches rather than close into a loop. The distance between the two foci is 2c, where c²=a²+b² (note: a plus, not a minus, which distinguishes this from the ellipse relationship c²=a²−b²). For x²/16 − y²/9 = 1: a=4, b=3, c=√(16+9)=5, so vertices at (±4,0) and foci at (±5,0) — always further out than the vertices, which matches the physical shape of the curve opening around them.

What do I need to know about Real-World Uses of Hyperbolas?

Hyperbolic paths appear in several real physical situations. An object on an unbound (escape) trajectory passing a gravitating body — like an interstellar object flying past the Sun with more than escape velocity — follows a hyperbolic orbit rather than the closed ellipse of a bound orbit, swinging past once and leaving forever. Hyperbolic mirrors and lenses are used in Cassegrain telescopes to fold a long focal length into a compact tube. LORAN and older radio navigation systems used hyperbolic positioning: the difference in arrival time of a signal from two fixed transmitters places the receiver on one branch of a hyperbola (since a constant difference in distance to two foci is the defining property of a hyperbola), and a second pair of transmitters gives a second hyperbola whose intersection pinpoints location — the same defining property that makes hyperbolas useful for sonic boom shockwave shapes and cooling tower silhouettes.

What should I know about Rectangular (Equilateral) Hyperbolas?

A rectangular hyperbola, also called an equilateral hyperbola, is the special case where the two asymptotes are perpendicular to each other — this happens precisely when a = b in the standard equation x²/a² - y²/b² = 1. In this special case, the asymptotes simplify to the lines y = x and y = -x, and if the hyperbola is instead rotated 45° and written in the simpler form xy = k, it becomes exactly the "inverse proportion" curve taught earlier in algebra (y = k/x). This connects two topics that often seem unrelated at first glance: the inverse-proportion graphs from basic algebra are a specific, rotated instance of the general hyperbola family studied later in conic sections, and recognizing xy = k as a rectangular hyperbola explains some of its symmetric, two-branch shape that a simple "y = k/x" description doesn't immediately make obvious.

Hyperbola Reference Table: Standard Forms and Key Features

The two standard orientations of a hyperbola centered at the origin share the same relationships between a, b, c, and eccentricity:

The key structural difference from an ellipse is the plus-versus-minus sign and the relation c² = a² + b² (rather than c² = a² − b² for an ellipse), which is also why a hyperbola's eccentricity always exceeds 1, while an ellipse's is always below 1.

Worked Example: Finding Foci, Asymptotes, and Eccentricity

Find the key features of the hyperbola x²/16 − y²/9 = 1. This is a horizontal hyperbola with a² = 16 (so a = 4) and b² = 9 (so b = 3). Foci: c² = a² + b² = 16 + 9 = 25, so c = 5. The foci sit at (±5, 0), vertices at (±4, 0). Asymptotes: y = ±(b/a)x = ±(3/4)x — these are the two straight lines the hyperbola branches approach but never touch as x → ±∞. Eccentricity: e = c/a = 5/4 = 1.25. Since e is only slightly above 1, this hyperbola's branches are relatively "wide" and open rather than sharply pinched; a much larger e (say 4 or 5) would describe a hyperbola whose branches are narrow and almost parallel to their asymptotes right from the vertex. As a quick verification, every point on this curve must satisfy the focal-distance definition: the difference between distances to the two foci (±5,0) from any point on the hyperbola equals 2a = 8 exactly — you can check this at the vertex (4,0): distance to (5,0) is 1, distance to (−5,0) is 9, and 9 − 1 = 8 ✓.

Rectangular Hyperbolas and the Relationship xy = k

A special case called the rectangular (or equilateral) hyperbola has perpendicular asymptotes and can be written in the simple rotated form xy = k, where k is a nonzero constant — this is the same shape you see in an inverse proportionality relationship, like pressure vs. volume at constant temperature (Boyle's Law). For xy = 12, if x = 3, then y = 12/3 = 4; if x = 6, then y = 12/6 = 2; if x = −4, then y = 12/−4 = −3. Plotting these points reveals two branches: one in the first quadrant (both x and y positive) and one in the third quadrant (both negative), approaching the x-axis and y-axis as asymptotes without ever touching them. This form is "rectangular" because its asymptotes (the x-axis and y-axis here) are perpendicular to each other, unlike the standard-form hyperbola's asymptotes, which cross at an angle determined by a/b. Every xy = k hyperbola can in fact be rotated 45° to match the standard form x²/a² − y²/b² = 1 with a = b = √(2k) (for positive k) — the two descriptions represent the same curve, just in different coordinate orientations.