When it comes to an ellipse, I know that an ellipse with major axis $2a$ and minor axis $2b$ and a distance from the center to the foci $c$ has a relationship $b^2+c^2=a^2$.
This can be shown by using the definition of the focal radii and just pick the covertex to come up with this equality.
A similar situation with a hyperbola gives us $a^2+b^2=c^2$, but I don't know why this is true.
I am trying to convince my students with basic precalc material, can someone help me out ?