How to find the length of major and minor axis of ellipse given the length of two conjugate diameters and the angle between them?
I am aware about how to construct the ellipse using the above given facts(not by Rytz's Construction). I would like to know, independent of what method of construction one uses, how one can find the length of the axes.
Best Answer
Here's a geometric construction: if $MN$ and $DE$ are conjugate diameters, draw line $QQ'$ through $N$ perpendicular to $DE$ (see diagram below). Points $Q$ and $Q'$ must be chosen such that $NQ=NQ'=OD$. Major axis $IR$ is the bisector of angle $\angle QOQ'$ and minor axis $TS$ is perpendicular to it. Their lengths can be computed from: $$ \tag{1} IR=OQ'+OQ,\quad TS=OQ'-OQ. $$
If $ON=a$, $OD=b$ and the angle between them is $\theta$, then from the cosine rule applied to triangles $ONQ$ and $ONQ'$ we get: $$ OQ^2=a^2+b^2-2ab\sin\theta,\quad OQ'^2=a^2+b^2+2ab\sin\theta. $$ Inserting these into $(1)$ we finally obtain: $$ OR\cdot OS=ab\sin\theta,\quad OR^2+OS^2=a^2+b^2. $$ These equalities could have been directly derived, as they are well known properties of conjugate diameters (see properties 1. and 2. listed here).