I find that for all conic sections centered at origin, the distance of focus from centre is given as $ \sqrt{v-u}$, where the general conic is equation is $\frac{x^2}{u} + \frac{y^2}{v}=1$, with $v>u$
Illustrations
For circle $v=u$ and focus is at origin itself
For ellipse, it automatically agrees $\sqrt{v-u}$, when $v>u$
For hyperbola of the form $\frac{y^2}{v}+\frac{x^2}{(-u)} \ =1$ with $v>u$, we find the focal length as $\sqrt{v+u}$ (explanation: $(v-(-u) ) = v+u$)
The question
Why is there such a similarity? I know all of these are conic sections. So, I guess something related to that but I can't figure out anything concrete for it. I know how to derive the formula for each instance case, but what exactly is it which is unifying the cases?
This thinking seems to fail for the parabola, the quick reason I can give is that for a parabola, we only have one parameter to control it's shape other than it's centre.
I saw this post but the only answer is a vomit of algebra without any explanation for why the it is geometrically. Maybe there is a way to see this using projective geometry?
