I have recently learned about the Hyperbolas and it's properties. The formula for finding the focal length that is $$f^2 = a^2 + b^2$$, where f, a, and b is the focal length, distance from center to vertices, and distance from center to the co-vertices, respectively, is confusing to me that from where does this formula came from? and how is it derived? I want to know the proof of it. I searched in different books and even tried myself but I was unable to prove.
Focal length: the distance from the center to the focus of the hyperbola. Co-vertex: The endpoints of the conjugate axis of the hyperbola.
My brain is a kind of skeptical, it just doesn't accept things the way they are, so please help me.
Thanks!
We know that the focal lenght is the ditance of the foci from the origin. So we draw a circunference with centre $O(0,0)$ and radius $\sqrt{a^2+b^2}$.
From the graph, it easy to see that the radius of the circle is also the focal lenght and so: $$f^2=a^2+b^2$$