In the app below you can explore how a single construction can create ellipses, hyperbolas and circles.[br][br]Place point [math]F[/math] where you want and start the animation, or move point [math]A[/math] along the circle of center [math]O[/math] and radius [math]r[/math].[br][br]The position of [math]F[/math] relative to the circle determines the conic envelope:[br][list][*][math]F[/math] [color=#1e84cc][i]inside [/i][/color]→ [color=#1e84cc][i]ellipse [/i][/color]with foci [math]O,F[/math], and major semi-axis length [math]\frac{r}{2}[/math][/*][*][math]F[/math] [color=#1e84cc][i]coinciding [/i][/color]with [math]O[/math] → [color=#1e84cc][i]circle [/i][/color]with center [math]F\equiv O[/math] and radius [math]\frac{r}{2}[/math][/*][*][color=#1e84cc][i][math]F[/math] outside [/i][/color]→ [color=#1e84cc][i]hyperbola [/i][/color]with foci [math]F,O[/math] and semi-axis length [math]\frac{r}{2}[/math][br][/*][/list][*] [/*]