The curves ax2+by2=1  and Ax2+By2=1  intersect orthogonally, then

ax2+by2=1,  Ax2+By2=1                 ax2+by2=Ax2+By2                 ⇒(a−A)x2=(B−b)y2⇒x2y2=B−ba−A                 ax2+by2=1⇒dydx=−axby=m1                 Ax2+By2=1⇒dydx=−AxBy=m2                 m1×m2=−1⇒a.Ab.B.x2y2=−1           ⇒a.Ab.B(B−ba−A)=−1 ⇒1b−1B=1a−1A