In this figure, drag a focus and a vertex to create ellipses and hyperbolas.[br] [math]c[/math] is the center-to-focus distance.[br] [math]a[/math] is the center-to-vertex distance.[br]The [b]eccentricity [/b]of an ellipse or a hyperbola is defined as [math]\frac{c}{a}[/math].[br]The equation [math]\frac{x^2}{a^2}+\frac{y^2}{a^2-c^2}=1[/math] describes both ellipses and hyperbolas in the Cartesian plane centered at the origin with foci on the [math]x[/math]-axis. The shape is an ellipse if [math]a^2>c^2[/math] and the shape is a hyperbola if [math]a^2 \lt c^2[/math].[br]The [b]directrices [/b]of ellipses and hyperbolas are lines perpendicular to the focal axis, a distance [math]\frac{a^2}{c}[/math] from the origin.[br]The focus-directrix equation states that [math]PF=e\cdot PD[/math] for all conics. See if you can demonstrate this equation with the interactive figure here.

[i]This applet was developed for use with [url=https://www.pearson.com/en-us/subject-catalog/p/interactive-calculus-early-transcendentals-single-variable/P200000009666]Interactive Calculus[/url], published by Pearson.[/i]