Construction of major and minor axes of an ellipse given only 2 focus points($F_1,F_2$) and a point $P$ that is on the ellipse.

Question

Construction of major and minor axes of an ellipse given only 2 focus points $(F_1,F_2)$ and a point $P$ that is on the ellipse.

Suppose we define $|F_1P|+|PF_2|:=l$

First I constructed the perpendicular bisector of $F_1F_2$ this is essentially the minor axis given the symmetry of the foci.

To obtain the boundaries, I was thinking of taking the segment $F_1P$ and extend it (Euclid's 2nd Axiom). Using a compass, with radius set to $|PF_2|$ and centred at $P$, draw a circle and mark the intersection with the extended line $F_1P$ as $F_1'$ where this point is on the opposite side of $F_1$ with respect to $P$. Now I construct the perpendicular bisection of $F_1F_1'$ to determine its midpoint (let's call it $M$).

Am I wrong to say that $M$ lies on both the minor axis as well as the ellipse?

Answer 1

You're close. $F_1M$ is half the major axis, which matches the length $F_1W$ where $W$ is on the minor axis. But the direction from $F_1$ to $M$ may not be what you need.

Center your compasses at $F_1$ and draw a circle through $M$. This intersects the $F_1F_2$ bisector at points $W_1$ and $W_2$ where the ellipse intersects the minor axis.

Keep the same radius and now center on the already constructed midpoint of $F_1F_2$. This circle identifies the vertices $V_1$ and $V_2$ on the major axis.