Two conic sections are given, whose directrices are perpendicular to each other. It is known that $A$, $B$, $C$, $D$ are intersections of the conics. Prove that $A$, $B$, $C$, $D$ lie on a common circle.
I don't know whether it's well-known. I have been trying some ways but most of them is helpless. I am thinking of using coordinates but I don't know how to write the equation of conics when we got the directrix.
What should I do now?
I will consider the case of two ellipses $E_1$, $E_2$, the first with a horizontal major axis, the second with a vertical major axis of equation $$E_i :P_i(x,y)= \frac{x^2}{a_i^2} + \frac{y^2}{b_i^2} + L_i(x,y) = 0$$ $i=1,2$, where $L_i$ are linear forms and $a_1>b_1$, $a_2 < b_2$.
The common points satisfy any linear combination $$P(x,y) = \lambda P_1(x,y) + (1-\lambda) P_2(x,y)=0$$
We can choose $\lambda\in (0,1)$ so that the coefficients of $x^2$, $y^2$ are equal. That will be the equation of a circle.