The curves $a x^2+b y^2=1$ and $A x^2+B y^2=1$ intersect orthogonally, then

$\frac{1}{a}-\frac{1}{A}=\frac{1}{b}-\frac{1}{B}$

We have seen that the curves $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $\frac{x^2}{c^2}+\frac{y^2}{d^2}=1$ intersect orthogonally, if $a^2-b^2=c^2-d^2$.

So, given curve will intersect orthogonally, if $\frac{1}{a}-\frac{1}{b}=\frac{1}{A}-\frac{1}{B}$.