To define an ellipse in terms of a pair of points that lie on its minor axis

Question

Just as an ellipse can be defined in terms of the sum of distances from its foci, that are both points on the major axis, can any ellipse be defined with reference to a pair of suitably chosen points on its minor axis? If so, how?

Answer 1

Yes. Consider the ellipse $x^2/a^2+y^2/b^2=1$, with the points $P=(0,\frac{-b}2)$ and $Q=(0,\frac{b}2)$ on the minor axis. Let $X=(x,y)$ be a point on the ellipse.

Then: \begin{align} PX^2 &= x^2+(y+b/2)^2\\ PY^2 &=x^2+(y-b/2)^2\\ PX^2-PY^2 &= 2by\\ \frac{PX^2+PY^2 -b^2\!/2}{2a^2} &= \frac{x^2}{a^2} + \frac{y^2}{a^2}\\ \end{align}

So the equation of the ellipse is $$\frac{PX^2+PY^2-b^2\!/2}{2a^2} + \frac{(PX^2-PY^2)^2}{4b^2}\left(\frac{1}{b^2}-\frac{1}{a^2}\right) = 1$$