Asymptotes to Rectangular Hyperbola

Theorem

Let $\KK$ be a rectangular hyperbola.

The asymptotes of $\KK$ are perpendicular.

Proof

Let $\KK$ be embedded in a cartesian plane in reduced form with the equation:

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

From Asymptotes to Hyperbola in Reduced Form, the asymptotes of $\KK$ can be expressed in the form:

$y = \pm \dfrac b a x$

By definition of a rectangular hyperbola, $\KK$ is such that:

$a = b$

Hence the asymptotes of $\KK$ can be expressed in the form:

$y = \pm x$

The slope of $y = x$ is $1$.

The slope of $y = -x$ is $-1$.

We have that:

$1 = - \dfrac 1 {-1}$

Hence the result, from Condition for Straight Lines in Plane to be Perpendicular: Slope Form.

$\blacksquare$

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