I was recently introduced to the world of conic sections, and I came across two different “definitions” for an ellipse and a hyperbola.

I was wondering if there was an easy way to prove that they happen to be “equivalent”.

First, it was established that:

Given a line $r$, a point $F$ not in $r$, and $e in mathbb R^+_*$, an ellipse/hyperbola is the locus of the points $P in mathbb R^2$ such that $overline{PF} = eoverline{Pr}$ (where $e<1$ for the ellipse and $e>1$ for the hyperbola)

What I would like to verify is that the set of points $bigg{(x,y) in mathbb R^2: dfrac{x^2}{a^2} pm dfrac{y^2}{b^2} =1bigg}$ satisfy the definition given.

Any thoughts?

Thanks in advance.