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.
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This should be tagged as conic-sections, geometry instead of algebraic-geometry. – user_of_math Oct 06 '14 at 04:40
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Are $a$ and $b$ simply semi major and minor axes? – Kaster Oct 06 '14 at 04:43
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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 $c^2 = a^2 e^2 = a^2 (1+ \frac{b^2}{a^2}) = a^2 + b^2$.
James Harrison
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That and the linear eccentricity $c$ are defined/explained here: https://en.wikipedia.org/wiki/Conic_section#Features – James Harrison Oct 06 '14 at 22:06