Ellipse and Hyperbola — Standard Forms

The ellipse, with eccentricity $0 < e < 1$, has standard equation (major axis along the $x$-axis, centre at the origin):

Here $a$ is the semi-major axis and $b$ the semi-minor axis. The foci lie on the major axis at $(\pm c, 0)$, where $c^2 = a^2 - b^2$, and the eccentricity is $e = c/a$ (always less than $1$ since $c<a$).

The hyperbola, with eccentricity $e > 1$, has standard equation (transverse axis along the $x$-axis, centre at the origin):

Here the foci lie at $(\pm c, 0)$, but now $c^2 = a^2 + b^2$ (a $+$, not a $-$, distinguishing it from the ellipse relation), and $e = c/a$ is always greater than $1$ since $c > a$.

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