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Given that the definition of an ellipse is the set of points (x,y) where the distance from focus 1 to (x,y) plus the distance from from focus 2 to (x,y) is a constant, and given that constant, and the positions of the foci, how can I find the lengths of the major and minor axes of the ellipse? I'm struggling to relate these terms. Thanks for your help!

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    $\begingroup$ If $S, S'$ are foci, and $P$ is a point on the ellipse, then $SP+SP' = 2a$ and $SS' = 2ae$, where $e$ is the eccentricity of the ellipse. From these find $e$ and $b^2 = a^2(1-e^2)$ $\endgroup$ – user348749 Sep 11 '16 at 2:32
  • $\begingroup$ Can you explain your notation? What is P' ? $\endgroup$ – user1917407 Sep 11 '16 at 2:40
  • $\begingroup$ Sorry, it should be $SP+S'P = 2a$ $\endgroup$ – user348749 Sep 11 '16 at 2:43
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For convenience, I'll call the distance between the foci $\ell$.

The major axis goes through both foci; if you have the length constant $d$, the two lines it makes covers the distance between the foci once and the distance from one focus to the end of the ellipse twice. --Which actually means that it covers the semimajor axis (which is distance from center to end, passing through the focus) twice, so the major axis has length of exactly $d$.

The minor axis is slightly harder, being the perpendicular bisector of the major axis. The extreme point is distance $d/2$ from each focus, and is the hypotenuse of a right triangle that has one base that is the distance between one focus and the center $\ell/2$. ...conveniently, this means that the semiminor axis has length $\sqrt{(d/2)^2-(\ell/2)^2} = \sqrt{d^2-\ell^2}/2$, so the minor axis has length $\sqrt{d^2-\ell^2}$.

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Ellipse property

$$ b^2 = a^2 - c^2 $$

$$ (2b)^2 = (2a)^2 - (2c)^2 $$

Note that since

$$ majoraxis=2a= d_1+d_2 ,$$

$$[minoraxis^2 =(2 a)^2 - (inter-focal distance)^2 ]$$

So

$$ minoraxis^2 = (d_1+d_2)^2 - (inter-focal distance)^2 $$

$$ minoraxis = \sqrt{(d_1+d_2)^2 - (inter-focal distance)^2}.$$

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