Further coordinate systems

Ellipse and hyperbola

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is an ellipse with $b^2=a^2(1-e^2)$; $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is a hyperbola with $b^2=a^2(e^2-1)$. Both have foci $(\pm ae,0)$ and directrices $x=\pm\frac ae$; distance to a focus is $e$ times distance to the matching directrix.