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Rectangular Hyperbola — Definition, Formula & Examples

A rectangular hyperbola is a hyperbola whose two asymptotes are perpendicular to each other. This happens when the semi-transverse axis and semi-conjugate axis have equal length.

A rectangular (or equilateral) hyperbola is a hyperbola for which a=ba = b, where aa is the semi-transverse axis and bb is the semi-conjugate axis. Its asymptotes intersect at right angles, and in standard position centered at the origin the equation reduces to x2−y2=a2x^2 - y^2 = a^2. When rotated 45°, the equation takes the form xy=cxy = c for some constant cc.

Key Formula

x2−y2=a2x^2 - y^2 = a^2
Where:
  • xx = Horizontal coordinate
  • yy = Vertical coordinate
  • aa = Equal semi-transverse and semi-conjugate axis length

How It Works

In the general hyperbola equation x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, the asymptotes are y=±baxy = \pm\frac{b}{a}x. Setting a=ba = b makes the asymptote slopes +1+1 and −1-1, which are perpendicular. The eccentricity of every rectangular hyperbola is e=2e = \sqrt{2}. If you rotate the axes by 45°, the equation simplifies to xy=kxy = k, which is the familiar inverse-variation curve you encounter in algebra.

Worked Example

Problem: Determine whether the hyperbola x2−y2=16x^2 - y^2 = 16 is rectangular, and find its asymptotes and eccentricity.
Rewrite in standard form: Divide both sides by 16 to get the standard hyperbola form.
x216−y216=1\frac{x^2}{16} - \frac{y^2}{16} = 1
Identify a and b: Here a2=16a^2 = 16 and b2=16b^2 = 16, so a=4a = 4 and b=4b = 4. Since a=ba = b, this is a rectangular hyperbola.
a=b=4a = b = 4
Find asymptotes and eccentricity: The asymptotes are y=±bax=±xy = \pm\frac{b}{a}x = \pm x. These lines are perpendicular. The eccentricity is e=1+b2a2=2e = \sqrt{1 + \frac{b^2}{a^2}} = \sqrt{2}.
y=x,y=−x,e=2y = x, \quad y = -x, \quad e = \sqrt{2}
Answer: Yes, x2−y2=16x^2 - y^2 = 16 is a rectangular hyperbola with asymptotes y=xy = x and y=−xy = -x, and eccentricity 2\sqrt{2}.

Why It Matters

The rotated form xy=kxy = k models inverse variation, which appears in physics (Boyle's law: pressure times volume equals a constant) and economics (unit price times quantity equals fixed revenue). Recognizing this curve as a conic section connects algebra concepts to real-world relationships.

Common Mistakes

Mistake: Assuming every hyperbola with perpendicular-looking asymptotes is rectangular.
Correction: Always verify that a=ba = b in the standard equation. If the graph axes are scaled unequally, asymptotes can appear perpendicular even when they are not.