Rectangular Hyperbola: xy = c² Properties & Graphs

A rectangular hyperbola is the curve given by the equation xy = c², where c is any non-zero constant. Its graph has two separate branches: one in the first quadrant where both x and y are positive, and one in the third quadrant where both are negative. Each branch comes closer and closer to the coordinate axes as it stretches outward, but it never actually touches them.

This curve matters because it is the graph of every inverse proportion you meet in science. In physics, Boyle's law says pressure times volume stays constant at fixed temperature, so a pressure-volume graph is a rectangular hyperbola. The same shape appears in economics and chemistry, and because it is the simplest hyperbola of all, mastering it first makes every other conic section easier to understand.

Key Concepts

• Asymptotes: the coordinate axes x = 0 and y = 0. The branches approach these lines forever without meeting them.
• Vertices: the points (c, c) and (−c, −c), the points of each branch nearest the origin.
• Transverse axis: the line y = x joining the two vertices; its length is 2√2·c.
• Eccentricity: exactly e = √2 for every rectangular hyperbola, fixed for all of them, unlike general hyperbolas.
• Foci: the points (c√2, c√2) and (−c√2, −c√2), lying on the transverse axis.
• Parametric form: every point can be written as (ct, c/t) for some non-zero number t, since (ct)(c/t) = c².

Formulas

• Equation: xy = c², with c a non-zero real constant.
• Eccentricity: e = √2 (exact, always).
• Transverse axis length: 2√2·c. The conjugate axis has the same length, which makes the asymptotes perpendicular and earns the name rectangular.
• Parametric equations: x = ct, y = c/t, with t ≠ 0; the latus rectum also measures 2√2·c.
• Assumptions: x and y are real numbers and c ≠ 0. In pure geometry there are no SI units; in applications the units multiply, for example pascal·metre³ for pressure × volume.
• Limits: the formula says nothing on the axes themselves, since setting x = 0 would demand 0 = c², which is impossible.

Worked Example

Take xy = 16, so c² = 16 and c = 4. Step 1: the vertices are (4, 4) and (−4, −4). Step 2: transverse axis length = 2√2 × 4 = 8√2 ≈ 8 × 1.414 = 11.31. Step 3: the foci are (4√2, 4√2) ≈ (5.66, 5.66) and (−5.66, −5.66). Step 4: pick t = 2 in the parametric form to get the point (8, 2), and check 8 × 2 = 16. Step 5: pick t = 0.5 to get (2, 8), and check 2 × 8 = 16. Step 6: the first-quadrant branch falls steeply through (2, 8) and the vertex (4, 4), then flattens past (8, 2), never touching the axes. The third-quadrant branch is its mirror image through the origin.

Common Misconceptions

• "If you travel far enough along a branch, it touches the axis." Wrong. From xy = c², setting x = 0 would require 0 = c², impossible for c ≠ 0. The axes are asymptotes: approached forever, never reached.
• "Rectangular means the curve looks like a rectangle." Wrong. It means the two asymptotes are perpendicular, meeting at a right angle; equivalently a = b in x²/a² − y²/b² = 1.
• "Every hyperbola has eccentricity √2." Wrong. Only rectangular hyperbolas do; a general hyperbola has e > 1, and its value changes with the shape.

Real-World Applications

• In physics, every Boyle's law isotherm (pressure vs volume at constant temperature) is a rectangular hyperbola.
• In economics, a demand curve with constant total spending (price × quantity = fixed budget) has exactly this shape.
• In the lab, any experiment with two inversely proportional quantities, such as time taken versus number of workers on a fixed job, traces this curve.

A rectangular hyperbola is the curve xy = c²: two branches in opposite quadrants, the axes as asymptotes, vertices at (c, c) and (−c, −c), a fixed eccentricity of √2, and the parametric form (ct, c/t). It is the picture of inverse proportion, and the foundation for understanding all hyperbolas.