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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?

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  • $\begingroup$ Boundaries? You are asked for the major and minor axes. The major axis is the line through the foci and the minor axis is the perpendicular bisector of that line. You do not need $P$. $\endgroup$ – almagest Mar 22 '16 at 10:19
  • $\begingroup$ Yeah that's my initial impression of the question. but I think that they want the bounds for the ellipse on the major and minor axis though it's not explicitly stated. I guess it's a problem of bad phrasing? $\endgroup$ – Danxe Mar 22 '16 at 10:26
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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.

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