Answer
$a=b$
Work Step by Step
Consider the hyperbola centered in origin:
$\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$
The asymptotes of this hyperbola are:
$y-0\pm\dfrac{b}{a}(x-h)$
$y=\pm\dfrac{b}{a}x$
In order to have perpendicular asymptotes we must have:
$-\dfrac{b}{a}\cdot\dfrac{b}{a}=-1$
$b^2=a^2$
$b=a$
So the hyperbolas with perpendicular asymptotes can be written:
$\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$
This also works in the general case when the hyperbola is centered in $h,k)$.
$\dfrac{(x-h)^2}{a^2}-\dfrac{(y-k)^2}{a^2}=1$
