Geometric or analytic proof that in hyperbola, $c^2=a^2+b^2$

Question

How to prove (geometric or analytic) that in hyperbola $c^2=a^2+b^2$? Given that $a$ is the undirected distance of the center to one of the vertices, $b$ is the undirected distance of one of the endpoints of the conjugate axis and $c$ is the undirected distance from the center to one of the foci.

Answer 1

Note that the eccentricity of a hyperbola is defined as $e = \sqrt{1+\frac{b^2}{a^2}}$. You know that $c = ae$ (by definition), therefore $c^2 = a^2 e^2 = a^2 (1+ \frac{b^2}{a^2}) = a^2 + b^2$.