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

$a{x}^2+b{y}^2=1,A{x}^2+B{y}^2=1$

                $a{x}^2+b{y}^2=A{x}^2+B{y}^2$

                $\Rightarrow \left(a-A\right)x^2=\left(B-b\right)y^2\Rightarrow \frac{x^2}{y^2}=\frac{B-b}{a-A}$

                $a{x}^2+b{y}^2=1\Rightarrow \frac{dy}{dx}=\frac{-ax}{by}=m_1$

                $A{x}^2+B{y}^2=1\Rightarrow \frac{dy}{dx}=\frac{-Ax}{By}=m_2$

                $m_1\times m_2=-1\Rightarrow \frac{a.A}{b.B}.\frac{x^2}{y^2}=-1$

          $\Rightarrow \frac{a.A}{b.B}\left(\frac{B-b}{a-A}\right)=-1\Rightarrow \frac{1}{b}-\frac{1}{B}=\frac{1}{a}-\frac{1}{A}$