Ejemplo 4. Cónicas por envolventes

[i]Dada una circunferencia y un punto P interior, que no sea el centro. Sea A un punto cualquiera de la circunferencia y r la recta perpendicular al segmento PA por el punto A.[/i][br][br][i]Hallar el lugar geométrico que determina la recta r cuando A recorre la circunferencia.[/i][br][br][i]Averigua qué pasa cuando el punto P es exterior a la circunferencia.[/i][br][br]Al igual que en ejemplos anteriores, comenzamos creando los objetos iniciales: circunferencia y los puntos A en la circunferencia y P distinto del centro.[br][br][img width=188,height=170]data:image/png;base64,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[/img][br][br]A continuación trazamos el segmento AP y la recta perpendicular por el punto A.[br][br][img width=288,height=201]data:image/png;base64,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deseamos obtener el lugar geométrico de una recta, no es posible utilizar la herramienta [b]Lugar geométrico[/b] que solo se puede aplicar sobre puntos. Por tanto, la alternativa será activar el rastro de la recta y posteriormente, mover el punto A de forma manual o activando la animación automática.[br][br]El resultado será una elipse obtenida como envolventes.[br]
Cónicas por envolventes
Para comprobar qué ocurre cuando P es exterior a la circunferencia, movemos el punto P, pulsando las teclas [b]Ctrl-F[/b] antes de volver a pulsar sobre [img width=20,height=23]data:image/png;base64,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[/img] para iniciar la animación.[br]

Information: Ejemplo 4. Cónicas por envolventes