Ejemplo 13. Podarias

A partir de una curva c se traza la recta tangente en un punto P de c.      [br][br]Sea Q el punto de intersección de la recta tangente anterior con la recta perpendicular a la tangente por un punto O.[br][br]Se denomina podaría de la curva c al lugar geométrico del punto Q cuando P recorre la curva c. [br][br]Obtener la podaría de la elipse cuando O es el centro.[br][br]Comenzamos dibujando la elipse a partir de los focos A y B, y de un punto de la elipse C,[br]en la que creamos un nuevo punto P.[br][br][img width=261,height=142]data:image/png;base64,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[/img][br][br]Utilizamos el comando [b]Centro(cónica)[/b] para obtener el centro de la elipse al que cambiamos el nombre para denominarlo O.[br][br][img 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Vt8RRniTTZx7pJGf5tR9s11Pv6RbSf9crz/AOKQVq0UAZI1TVs4Ph6f6rcxH/2al/tXUf8AoAXf/f2L/wCKrVooAyTqmp4yvh+5P1nhH/s1A1DWX+7oYT/rrdoP5A1rUUAZW/xFJ0g02D6zSSf+yrTTZa7N/rdYhgHpbWg/m5ateigDJXw+j/8AH3qOoXWeqtcFF/JNtWrXSdOsjm2soI267gg3fn1q5RQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQB/9k=[/img][br][br]Utilizando la herramienta [b]Tangentes[/b], trazamos la recta tangente a la elipse por el punto P.[br][br][img 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con la herramienta [b]Recta Perpendicular[/b] trazamos la recta perpendicular a la recta anterior por el punto O, encontrando el punto Q intersección de las dos rectas.[br][br][img width=343,height=167]data:image/png;base64,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podaría es el lugar geométrico generado por el punto Q al mover P por la elipse.[br][br]El lugar se puede obtener aplicando la herramienta [b]Lugar[/b] sobre Q y P.[br][br]En el applet siguiente aparece la podaría de la elipse con respecto al centro.[br]
Podaria

Information: Ejemplo 13. Podarias