The medians AD and BE of the triangle with vertices A (0, b), B(0, 0) and C (a, 0) are mutually perpendicular if

Midpoint D of BC is (a/2, 0) and midpoint E of CA is (a/2, b/2)

Now slope of $AD=-2\frac{b}{a},$ slope of $BE=\frac{b}{a}$

$AD\perp BE\Rightarrow -\frac{2b}{a}.\frac{b}{a}=-1\Rightarrow 2b^2=a^2\Rightarrow a=\pm \sqrt{2}b$