I am currently (in my Pre-Calculus course) deriving the equations of the conic sections. I very much understand how the relationship, in an ellipse, between $a, b$, and $c$ is established. Knowing that in an ellipse, the sum of the distances from each focus to the point $(0, b)$, one endpoint of the minor axis, is equivalent to the sum of the distances from each focus to the right-hand vertex, by the very locus definition of the ellipse. Seeing this fact, I set the distance from $(0, b)$ to $(c, 0)$ equal to the distance from $(a, 0)$ to $(c, 0)$ and, through algebra, arrived at the Pythagorean relationship $b^2=a^2-c^2$. (*Note - this is based on an ellipse centered at the origin with major axis lying on the x-axis, but of course I understand how the relationship is maintained no matter how we translate and orient the ellipse. I just used this basic case for the derivation.)
However, I cannot say that I understand the Pythagorean relation established by the hyperbola. I began the derivation with a hyperbola centered at the origin and transverse axis lying on the x-axis. The point $(0, b)$ is not even on the hyperbola itself, while $(a, 0)$ and $(c, 0)$ are, being representative of the right-hand vertex and right-hand focus respectively; yet my textbook throws out the relationship $a^2+b^2=c^2$ out of nowhere and uses that relationship to finish the derivation of the equation. I am comfortable with all other parts of the derivation except this Pythagorean relationship. It comes quite literally completely out of nowhere to me, and I have searched the internet for hours today trying to find the proof of this relationship, and have found absolutely nothing. One site I went to, Purplemath, even wrote verbatim that the proof was "long and painful" and said to just "memorize" the relationship and "move on."
My ultimate question is thus: where on earth does this relationship, $a^2+b^2=c^2$, for a hyperbola, even come from? What is the geometric/algebraic reasoning? I want to fully understand this derivation and this is the sole hindrance.
Thanks very much!