How to construct the foci of an ellipse given both its axes' support lines and two points on the conic

I'm gonna rewrite the solution from Inteligenti pauca's link, which is more general than the problem I proposed as it considers the axes any two random conjugated diameters.

draw lines $AN$ and $BM$, both parallel to diameter 1, with $N$ and $M$ belonging to diameter 2. Let $B_1'$ be the reflection of $B$ w.r.t $M$ and let $A_1'$ be the reflection of $A$ w.r.t. $N_1$ (the intersection of parallel to diameter 2 through $A$ with diameter 1, not drawn above).

Now draw a line $MX \parallel AB$ through $M$, with $X \in AA_1'$ and draw $MY \parallel A_1'B_1'$ through $M$ with $Y \in AA_1'$.

Now take $C = AA_1' \cap BM$ and draw a perpendicular to diameter 2 through $N$ and let point $D$ lie on it such that $ND = \sqrt{CX \cdot CY}$. The circle $\odot (O,OD)$ meets diameter 2 in vertices $V_1$ and $V_2$ of the desired ellipse. To find the other vertices in my problem it is enough to do an dilation on the circle we found, but the rest of the solution works for any pair of conjugated diameters.