Equation of Rectangular Hyperbola in Reduced Form

Theorem

Let $\KK$ be a rectangular hyperbola whose transverse axis and conjugate axis are of length $2 a$.

Let $\KK$ be aligned in a cartesian plane in reduced form.

$\KK$ can be expressed by the equation:

$x^2 - y^2 = a^2$

Proof

From Equation of Hyperbola in Reduced Form in Cartesian Frame, a hyperbola can be expressed by the equation:

$\dfrac {x^2} {a^2} - \dfrac {y^2} {b^2} = 1$

For a rectangular hyperbola:

$a = b$

Hence $\KK$ can be expressed by the equation:

$\dfrac {x^2} {a^2} - \dfrac {y^2} {a^2} = 1$

Multiplying both sides by $a^2$

$x^2 - y^2 = a^2$

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): hyperbola
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): hyperbola