tag:blogger.com,1999:blog-42360889904502641302024-02-06T21:38:30.250-08:00APLICACIONES DE LA CÓNICAS A LA ARQUITECTURADaniel Espinozahttp://www.blogger.com/profile/01853802069444746502noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-4236088990450264130.post-1774097699413210512010-09-11T00:52:00.000-07:002010-09-11T00:52:48.830-07:00JANKARLO ALANIA VILCACHAGUA<div class="separator" style="clear: both; text-align: center;"><strong><span style="color: #660000; font-family: "Courier New", Courier, monospace; font-size: x-large;">CÓNICAS Y SU APLICACIÓN A LA ARQUITECTURA</span></strong></div><div class="separator" style="clear: both; text-align: center;"><br />
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</div><div class="separator" style="clear: both; text-align: center;"><span style="font-family: "Courier New", Courier, monospace; font-size: large;"><strong>CÓNICAS</strong></span></div><div class="separator" style="clear: both; text-align: center;"><br />
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</div><div class="separator" style="clear: both; text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;">Analizando la Historia de la humanidad principalmente la Historia del pensamiento en la antigua Grecia, se observa cómo los matemáticos y pensadores se han ocupado de analizar las formas óptimas en la geometría y en la naturaleza. </span></div><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;"></span></div><span style="font-family: "Courier New", Courier, monospace;"><div style="text-align: justify;"><br />
</div></span><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;">Quizá el descubrimiento más importante relacionado con uno de los grandes problemas de la geometría griega sea el que realizó MENECMO, matemático griego (350 a. de C.), intentando conseguir la duplicación del cubo (problema irracional: construir un cubo de doble volumen que otro dado): LAS CÓNICAS, curvas que se obtienen como secciones por medio de un plano de tres tipos de conos circulares rectos distintos según el ángulo del vértice fuese agudo, recto u obtuso.</span></div><span style="font-family: "Courier New", Courier, monospace;"><div style="text-align: justify;"><br />
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</div></span><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;">MENECMO descubre estas curvas como secciones de un cono circular recto por un plano perpendicular a una generatriz. </span></div><br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiAUamRlZZnsAGmrB-n806d5f88LqE3Y4sm9ChE4B6DKN1SUONx4jZQGpFnT3pRVY6e6rWGvrMrEaj8X1WDLeIC3JJg48ALbERXrY2qv3pN05W-dq8nwhQCNPpAbjTx6yQXwiMvjfF8ki0/s1600/A1.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiAUamRlZZnsAGmrB-n806d5f88LqE3Y4sm9ChE4B6DKN1SUONx4jZQGpFnT3pRVY6e6rWGvrMrEaj8X1WDLeIC3JJg48ALbERXrY2qv3pN05W-dq8nwhQCNPpAbjTx6yQXwiMvjfF8ki0/s320/A1.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><span style="font-family: "Courier New", Courier, monospace;">CONO RECTÁNGULO: Giro en torno a un cateto de triángulo rectángulo isósceles</span></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhO8x8bzFDwYuGTHl3WGWZdNEcJ36ecXJCaCugDM9EA702J76S69CNXX5_6L3Yrx2JVPg4QG0gomjbK7kpmZ-ezNlq_RveHsdK51e-yMpH5B1fsfbwGZgM1SiVnFdnYoFVC3Zflku9SRt8/s1600/A2.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhO8x8bzFDwYuGTHl3WGWZdNEcJ36ecXJCaCugDM9EA702J76S69CNXX5_6L3Yrx2JVPg4QG0gomjbK7kpmZ-ezNlq_RveHsdK51e-yMpH5B1fsfbwGZgM1SiVnFdnYoFVC3Zflku9SRt8/s320/A2.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><span style="font-family: "Courier New", Courier, monospace;">CONO ACUTÁNGULO: Giro en torno al cateto mayor de un triángulo rectángulo</span></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8ID4u1ksc3BeNTYCiUPW8L4FOPC3j79oMQgP9QTvi3JPEbZHz9PlJTYWlGIOd4FWR2T0zwTYsQXTBbUx1oxNhXRg8pJd1dyMJldBA39zoBm7zCHK6CvN5B2Bl-tTEkrsyX0vkLYsl-Uk/s320/A4.gif" /></div><div class="separator" style="clear: both; text-align: center;"><span style="font-family: "Courier New", Courier, monospace;">CONO OBTUSÁNGULO: Giro en torno al cateto menor de un triángulo rectángulo.</span></div><div class="separator" style="clear: both; text-align: center;"><br />
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<span style="font-family: "Courier New", Courier, monospace;">Las secciones propuestas por Menecmo serían:</span><br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgs14dnXyn2AjiR_9HQcZ0VkmzncfXjRVhZodRbNkQCLPgKMY26L83pJexWqwZzF094AhZLQaQwLDleFwS5Ba2D5idFfEHdLi6cSmQUV1CxV5AzVPoAcd_XFnFOBiDOkt17knp1wszR-6U/s1600/AA1.jpg" imageanchor="1" style="clear: left; cssfloat: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="150" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgs14dnXyn2AjiR_9HQcZ0VkmzncfXjRVhZodRbNkQCLPgKMY26L83pJexWqwZzF094AhZLQaQwLDleFwS5Ba2D5idFfEHdLi6cSmQUV1CxV5AzVPoAcd_XFnFOBiDOkt17knp1wszR-6U/s200/AA1.jpg" width="200" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg74OHrIxzg97B1XUPf9ECIQhTpBHfiQMArHLv3oqjajotlAi2StfwJ35JF83RcIh2RDqNxN79PyRP3dD4lTNTSaP0hFmHpjM_NfxO7aOK8-h26Xd_TgKW2-rl8AV3v_oevaNAyH71gNJ0/s1600/AA4.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="150" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg74OHrIxzg97B1XUPf9ECIQhTpBHfiQMArHLv3oqjajotlAi2StfwJ35JF83RcIh2RDqNxN79PyRP3dD4lTNTSaP0hFmHpjM_NfxO7aOK8-h26Xd_TgKW2-rl8AV3v_oevaNAyH71gNJ0/s200/AA4.gif" width="200" /></a></div><br />
<span style="font-family: "Courier New", Courier, monospace;">Secciones en un cono rectángulo </span><span style="font-family: "Courier New", Courier, monospace;">PRODUCEN UNA PARÁBOLA</span><br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNoGT6imMpezECOtcfteL-Ae-eigfa5jVaDz08-uCAYbWNxJtGce003C-AWHCVRJEM-kRFrc8GCVxSoiI-eNaO8EK3tF5VOn0GWlHXXqmG3gM6LYgE7BcF4UFw4Z_Fki3wmQVxcUL4tdg/s1600/AA2.jpg" imageanchor="1" style="clear: left; cssfloat: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="150" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNoGT6imMpezECOtcfteL-Ae-eigfa5jVaDz08-uCAYbWNxJtGce003C-AWHCVRJEM-kRFrc8GCVxSoiI-eNaO8EK3tF5VOn0GWlHXXqmG3gM6LYgE7BcF4UFw4Z_Fki3wmQVxcUL4tdg/s200/AA2.jpg" width="200" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZbwahViWEsc5S9dN53zdAx0dJ5Ad6pQO0jxT-levSPn4r7axaM7ALAYkG5C2r-TJpVljjNqDTUYDsiP8XKTlc6JDAjPQjgxtZkkQf7J4H0vH5NLQsD574tw-RNPhm37_1aAkn_YfqrF0/s1600/AA5.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="150" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZbwahViWEsc5S9dN53zdAx0dJ5Ad6pQO0jxT-levSPn4r7axaM7ALAYkG5C2r-TJpVljjNqDTUYDsiP8XKTlc6JDAjPQjgxtZkkQf7J4H0vH5NLQsD574tw-RNPhm37_1aAkn_YfqrF0/s200/AA5.gif" width="200" /></a></div><br />
<span style="font-family: "Courier New", Courier, monospace;">Secciones en un cono acutángulo PRODUCEN UNA ELIPSE </span><br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMWefKDpPNJKDvg4ps5t35j46153zGKWg5lq5P8_vEVzxYaG0PoTQvqlNJusVwMjr68IJkJt4tK16L6Fp-Afzj3NmA0HA5XOAEu9rzF8gJ94pgXgN9yvvrzPWHmCLbJYfEy01XzBld-0k/s1600/AA3.jpg" imageanchor="1" style="clear: left; cssfloat: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="150" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMWefKDpPNJKDvg4ps5t35j46153zGKWg5lq5P8_vEVzxYaG0PoTQvqlNJusVwMjr68IJkJt4tK16L6Fp-Afzj3NmA0HA5XOAEu9rzF8gJ94pgXgN9yvvrzPWHmCLbJYfEy01XzBld-0k/s200/AA3.jpg" width="200" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhv0jUPj0iFovngZn0DXxSkXO3hk1EpcDkhPPfTxJScqSVHJcO5WoiulvIv9QJJUVQdelBjyZ3GN4zv_uIRXzHW2Pg-qRPO-ujWIDczQktGDktwvShUxpJYXlr1iVHavp8OsZgUvHBmg2c/s1600/AA6.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="150" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhv0jUPj0iFovngZn0DXxSkXO3hk1EpcDkhPPfTxJScqSVHJcO5WoiulvIv9QJJUVQdelBjyZ3GN4zv_uIRXzHW2Pg-qRPO-ujWIDczQktGDktwvShUxpJYXlr1iVHavp8OsZgUvHBmg2c/s200/AA6.gif" width="200" /></a></div><br />
<span style="font-family: "Courier New", Courier, monospace;">Secciones en un cono obtusángulo PRODUCEN UNA RAMA DE HIPÉRBOLE </span><br />
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<div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;">Fue APOLONIO de Perga (262-190 a. de C.) el primero en estudiarlas detalladamente y encontrar la propiedad plana que las definía. </span></div><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;"></span></div><span style="font-family: "Courier New", Courier, monospace;"><div style="text-align: justify;"><br />
</div></span><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;">APOLONIO, demostró por primera vez:</span></div><span style="font-family: "Courier New", Courier, monospace;"><div style="text-align: justify;"><br />
</div></span><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;">- que no es necesario considerar exclusivamente secciones perpendiculares a una generatriz del cono.</span></div><span style="font-family: "Courier New", Courier, monospace;"><div style="text-align: justify;"><br />
</div></span><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;">- que de un cono único pueden obtenerse los tres tipos de secciones cónicas sin más que variar la</span></div><span style="font-family: "Courier New", Courier, monospace;"><div style="text-align: justify;"><br />
</div></span><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;">inclinación del plano que corta al cono. </span></div><span style="font-family: "Courier New", Courier, monospace;"><div style="text-align: justify;"><br />
</div></span><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;">- que no es necesario sea el cono recto, es decir que el eje sea perpendicular al plano de la base circular.</span></div><span style="font-family: "Courier New", Courier, monospace;"><div style="text-align: justify;"><br />
</div></span><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;">- que puede sustituirse el cono de una hoja por el cono de dos hojas( par de conos orientados en sentido</span></div><span style="font-family: "Courier New", Courier, monospace;"><div style="text-align: justify;"><br />
</div></span><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;">opuesto, con vértices coincidentes y ejes sobre la misma recta. Lo que le lleva a descubrir que la</span></div><span style="font-family: "Courier New", Courier, monospace;"><div style="text-align: justify;"><br />
</div></span><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;">hipérbola ese una cónica con dos ramas.</span></div><br />
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</div><div class="separator" style="clear: both; text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;">Para Apolonio: Si una recta de longitud indefinida y que pasa siempre por un punto fijo se hace mover sobre la circunferencia de un círculo que no está en el mismo plano que el punto dado, de tal manera que pasa sucesivamente por todos los puntos de dicha circunferencia, entonces la recta móvil describirá la superficie de un cono doble recto si la recta el perpendicular al círculo u oblicuo si no lo es.</span></div><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;"></span></div><span style="font-family: "Courier New", Courier, monospace;"><div style="text-align: justify;"><br />
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</div></span><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;">Apolonio, dio el nombre a las curvas obtenidas mediante las secciones: </span></div><div style="text-align: justify;"><br />
</div><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;">ELIPSE: Resulta al inclinar el plano, sin llegar a</span></div><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;"></span></div><span style="font-family: "Courier New", Courier, monospace;"><div style="text-align: justify;"><br />
</div></span><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;">ser paralelo a ninguna de sus generatrices y sin llegar al ángulo que forma la generatriz del cono. </span></div><br />
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<span style="font-family: "Courier New", Courier, monospace;">PARÁBOLA: Resulta al cortar el cono con un plano paralelo a la generatriz del cono </span><br />
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<span style="font-family: "Courier New", Courier, monospace;">HIPÉRBOLA: Resulta, si el ángulo del plano es todavía mayor.</span> <br />
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</div><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;">Apolonio demostró también que las curvas cónicas tienen muchas propiedades interesantes, algunas de las cuales se utilizan actualmente para definirlas. Quizás las más interesantes y útiles que descubrió son las llamadas propiedades de reflexión de las cónicas: </span></div><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;"></span></div><span style="font-family: "Courier New", Courier, monospace;"><div style="text-align: justify;"><br />
</div></span><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;">1ª.- Reflexión de la parábola: Si se recibe luz de una fuente lejana con un espejo parabólico, de manera que los rayos incidentes son paralelos al eje del espejo, entonces la luz reflejada por el espejo se concentra en el foco. </span></div><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjMWNO1FqrIgJjKzEDCntNKn1DjPvMeLOu-vIpIWV7hxF24Vt_PsWQRDDdgD5LPRkfBQhvJQO3BQAlbKHu0dHRNrSkViD9f_zATqXcjTX8jwwBeZ16Z5Xf8yYdgVHjfE9uPin5FFrFHRPA/s1600/DD1.jpg" imageanchor="1" style="clear: left; cssfloat: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="127" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjMWNO1FqrIgJjKzEDCntNKn1DjPvMeLOu-vIpIWV7hxF24Vt_PsWQRDDdgD5LPRkfBQhvJQO3BQAlbKHu0dHRNrSkViD9f_zATqXcjTX8jwwBeZ16Z5Xf8yYdgVHjfE9uPin5FFrFHRPA/s200/DD1.jpg" width="200" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPYwc1eCy2EPG8l8-CjDqi0xTFpt9kcTfU1p7w-xj9gYtat8dRizHolee6MYDku_CGGX-tQO_oMKbd0oUVTfj2yynbrJXcHHmj0uJ1mR68GiBzsix7jezHse88O2grE0ZZc0xGopWVIBU/s1600/DD2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="144" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPYwc1eCy2EPG8l8-CjDqi0xTFpt9kcTfU1p7w-xj9gYtat8dRizHolee6MYDku_CGGX-tQO_oMKbd0oUVTfj2yynbrJXcHHmj0uJ1mR68GiBzsix7jezHse88O2grE0ZZc0xGopWVIBU/s200/DD2.jpg" width="200" /></a></div><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;">Existe la leyenda que dice: Arquímedes (287-212 a. de C.), ante el asedio de los romanos a la ciudad de Siracusa, utilizó esta propiedad de reflexión parabólica, (ideó un complejo sistema de espejos metálicos colocados en forma de parábola que concentraban los rayos solares sobre la flota romana) para incendiar las naves romanas. </span></div><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;"></span></div><span style="font-family: "Courier New", Courier, monospace;"><div style="text-align: justify;"><br />
</div></span><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;">En la actualidad esta propiedad se utiliza para los radares, las antenas de televisión, espejos solares. </span></div><span style="font-family: "Courier New", Courier, monospace;"><div style="text-align: justify;"><br />
</div></span><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;">2º.- Reflexión de la elipse: Apolonio demostró, que si se coloca una fuente de luz en el foco de un espejo elíptico, entonces la luz reflejada en el espejo se concentra en el otro foco. </span></div><br />
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<span style="font-family: "Courier New", Courier, monospace;">Hasta el siglo XVII, las cónicas eran conocidas y apreciadas a través de la obra de Apolonio.</span> <br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0mVlwY6ZqgsaQcoCQewIw4lMJqd_mU6x398FlwHmxC1Lv8rotKHMsZLaK00H0xRx04fRcMrEKm41TEfI_F3qz2eKXVWudcUkTg7WFzVk_Dp5t-4fOQWCRxJL-VJZtjASVJOXUcn-n7TE/s1600/EE2.jpg" imageanchor="1" style="clear: left; cssfloat: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0mVlwY6ZqgsaQcoCQewIw4lMJqd_mU6x398FlwHmxC1Lv8rotKHMsZLaK00H0xRx04fRcMrEKm41TEfI_F3qz2eKXVWudcUkTg7WFzVk_Dp5t-4fOQWCRxJL-VJZtjASVJOXUcn-n7TE/s320/EE2.jpg" /></a></div><span style="font-family: "Courier New", Courier, monospace;">DESCARTES (1596-1650), desarrolló un método </span><br />
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<span style="font-family: "Courier New", Courier, monospace;">para relacionar las curvas con ecuaciones, lo que </span><br />
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<span style="font-family: "Courier New", Courier, monospace;">dio origen a la Geometría Analítica. </span><br />
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<span style="font-family: "Courier New", Courier, monospace;">Las cónicas pueden representarse por ecuaciones</span><br />
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<span style="font-family: "Courier New", Courier, monospace;">cuadráticas en dos variables.</span><br />
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<span style="font-family: "Courier New", Courier, monospace;">El hecho que todas las ecuaciones cuadráticas</span><br />
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<span style="font-family: "Courier New", Courier, monospace;">representen secciones cónicas se lo debemos a </span><br />
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<span style="font-family: "Courier New", Courier, monospace;">Jan de Witt (1629-1672). </span><br />
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<div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;">Fue entonces cuando Galileo Galilei (1564-1642)</span></div><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;"></span></div><span style="font-family: "Courier New", Courier, monospace;"><div style="text-align: justify;"><br />
</div></span><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;">probó que los proyectiles se mueven según trayectorias parabólicas: </span></div><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;"></span> </div><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;"></span> </div><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;">El astrónomo Johannes Kepler (1571-1630)</span> </div><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;"></span></div><span style="font-family: "Courier New", Courier, monospace;"><div style="text-align: justify;"><br />
</div></span><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;">descubrió que las órbitas que describen los</span></div><span style="font-family: "Courier New", Courier, monospace;"><div style="text-align: justify;"><br />
</div></span><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;">planetas al girar alrededor del sol son elipses que tienen al sol en uno de sus focos.</span></div><div style="text-align: justify;"><br />
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</div><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;">Sin lugar a dudas las cónicas son las curvas más importantes que la geometría ofrece a la Física. No sólo a ella sino también al Arquitectura ya que con ellas se han logrado hacer verdaderas obras de arte.</span></div><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;"></span></div><span style="font-family: "Courier New", Courier, monospace;"><div style="text-align: justify;"><br />
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</div></span><div style="text-align: justify;"><span style="font-family: "Courier New", Courier, monospace;">Los invito a descubrirlas también en los objetos de la vida real y sobre todo en la arquitectura de los edificios.</span></div><br />
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<span style="font-family: "Courier New", Courier, monospace; font-size: large;"><strong>APLICACIONES DE LAS CÓNICAS EN LA ARQUITECTURA:</strong></span><br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjR52tY6x_VoFoleQ31shY0IW8F0TZgCE6kJ8VDCMujo-Eut8lwVbtR0dX8bBMI0LeNpMu2t38zpWmXCqgMkD2NJLSvqcdUPhlO1xGSbVudiyfhV5gNVgTG1FLMTIbGLTmPnOIWiy7Gno/s1600/PAR3.bmp" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="220" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjR52tY6x_VoFoleQ31shY0IW8F0TZgCE6kJ8VDCMujo-Eut8lwVbtR0dX8bBMI0LeNpMu2t38zpWmXCqgMkD2NJLSvqcdUPhlO1xGSbVudiyfhV5gNVgTG1FLMTIbGLTmPnOIWiy7Gno/s320/PAR3.bmp" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;">SANTIAGO CALATRAVA</div><div class="separator" style="clear: both; text-align: center;"><br />
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Daniel Espinozahttp://www.blogger.com/profile/01853802069444746502noreply@blogger.com2tag:blogger.com,1999:blog-4236088990450264130.post-49621703543517392542010-09-11T00:28:00.000-07:002010-09-11T00:31:28.943-07:00Coronado Leon Fernando<div style="text-align: center;"><span style="color: #073763; font-size: large;"><strong> <span style="color: #666666; font-size: x-large;">Aplicación de las cónicas en el Perú</span></strong></span></div><br />
No se puede pensar que solo obras de este tipo existen en las diversas partes del mundo, muy alejadas de nuestra realidad. Sino que aquí también en Perú, las encontramos:<br />
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<strong><span style="color: #073763; font-size: large;">Arquitectura Preincaica</span></strong><br />
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<span style="color: #073763; font-size: large;"><strong>Arquitectura Incaica</strong></span><br />
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<strong><span style="color: #073763; font-size: large;">Arquitectura Colonial</span></strong><br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRmBlEnAowBxX8xKRkMjIup2NmD5dlkLtOrqzpGlwGxanGpHDXbjz2e3AJXXiQ-FB-HVzi6Yq8P2hsZdRVHGl8HxrYA_z0VfWGfoDPiejpJSU7G989QU2CRzoxHN7iGLNSsjm6vcgJUcU/s1600/gonzalo+173.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRmBlEnAowBxX8xKRkMjIup2NmD5dlkLtOrqzpGlwGxanGpHDXbjz2e3AJXXiQ-FB-HVzi6Yq8P2hsZdRVHGl8HxrYA_z0VfWGfoDPiejpJSU7G989QU2CRzoxHN7iGLNSsjm6vcgJUcU/s320/gonzalo+173.jpg" /></a></div> Mirador de Yanahuara (Arequipa)<br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvUSWN5Few0pSIwNMHQMd9cTECE86yWIL7ysQbRcp-kEgfPtpPfE1HjPGkNNRk48QrUS6h-aD8q0O4s_pBWk68GG_fLmF4wJ0zdAIAkxOTa4n6EuwGo9-QCS2EP4ETH1RSqaIHOOcEFX0/s1600/arquitectura-claustros-compania.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvUSWN5Few0pSIwNMHQMd9cTECE86yWIL7ysQbRcp-kEgfPtpPfE1HjPGkNNRk48QrUS6h-aD8q0O4s_pBWk68GG_fLmF4wJ0zdAIAkxOTa4n6EuwGo9-QCS2EP4ETH1RSqaIHOOcEFX0/s320/arquitectura-claustros-compania.jpg" /></a></div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"> Claustros de la compañía</div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"> </div><br />
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<div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"><span style="color: #073763; font-size: large;"><strong>Arquitectura republicana</strong></span></div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"><br />
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</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3XAMJZ1S9x6T9d96FHKmUWJ_GFYVlufScgHkIdLJk8p-g0BTBBac6TDmurQSnKa9I0SnllrSbxZbX3CWQYRxKiAVAFTjRgUUeJz8nnUTDnMgzPK9fkQ91mxGp6lmBjQmvk9Eskabm9H4/s1600/interbank.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3XAMJZ1S9x6T9d96FHKmUWJ_GFYVlufScgHkIdLJk8p-g0BTBBac6TDmurQSnKa9I0SnllrSbxZbX3CWQYRxKiAVAFTjRgUUeJz8nnUTDnMgzPK9fkQ91mxGp6lmBjQmvk9Eskabm9H4/s320/interbank.jpg" /></a></div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"> Banco Interbank</div>Daniel Espinozahttp://www.blogger.com/profile/01853802069444746502noreply@blogger.com1tag:blogger.com,1999:blog-4236088990450264130.post-79218484806966193252010-09-10T18:45:00.000-07:002010-09-10T18:49:03.387-07:00DANIEL ESPINOZA CURI<div style="text-align: center;"><span style="font-size: x-large;">IMÁGENES de la APLICACIÓN de CÓNICAS en la ARQUITECTURA</span></div><div style="text-align: center;"><br />
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</div><div style="text-align: center;"><em><strong><span style="color: #0b5394;">Parábolas</span></strong></em></div><div style="text-align: center;"><br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxZgigEdc2WuundtDCkHGOZx7WUiFbagRYcbQEVH9kgqGUpM9ZRCVOFy-E_L-Xklgj6wqTFI5XT284aSrofeqqMoq3CLvrrSH7Ya7mVUvjVvmz_M5YHcTqyEaQCIywKC_iTNlV36Tawww/s1600/conicas6.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxZgigEdc2WuundtDCkHGOZx7WUiFbagRYcbQEVH9kgqGUpM9ZRCVOFy-E_L-Xklgj6wqTFI5XT284aSrofeqqMoq3CLvrrSH7Ya7mVUvjVvmz_M5YHcTqyEaQCIywKC_iTNlV36Tawww/s320/conicas6.jpg" /></a></div><div class="separator" style="clear: both; text-align: center;">Ciudad de las Artes y las Ciencias. Valencia.</div><div class="separator" style="clear: both; text-align: center;"><br />
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</div><div class="separator" style="clear: both; text-align: center;"><img border="0" height="300" src="http://www.catedu.es/matematicas_mundo/FOTOGRAFIAS/curvas9.jpg" width="400" /></div><div class="separator" style="clear: both; text-align: center;">Parábolas bajo el puente (La Manga del Mar Menor)</div><br />
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<div class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px; text-align: center;"><b><i><span style="color: #333399; font-family: Verdana; font-size: 9pt;">Hipérbolas </span></i></b></div><div align="center" class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px;"><br />
</div><div align="center" class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px;"><span style="color: #333399; font-family: Verdana; font-size: 8pt;"> <img border="0" height="274" src="http://www.catedu.es/matematicas_mundo/FOTOGRAFIAS/conicas22.jpg" width="400" /></span></div><div align="center" class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px;"><span style="color: #333399; font-family: Verdana; font-size: 8pt;">Central térmica (*)</span></div><div align="center" class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px;"><span style="color: #333399; font-family: Verdana; font-size: 8pt;"> </span></div><div align="center" class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px;"><span style="color: #333399; font-family: Verdana; font-size: 8pt;"> </span></div><div class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px; text-align: center;"><span style="color: #333399; font-family: Verdana; font-size: 9pt;"> <i><b>Elipses</b></i></span></div><br />
<div style="text-align: center;"><img border="0" height="266" src="http://www.catedu.es/matematicas_mundo/FOTOGRAFIAS/conicas9.jpg" width="399" /></div><div style="text-align: center;"><span style="color: #333399; font-family: Verdana; font-size: x-small;">Anfiteatro de Pompeya </span></div><div align="center" class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px;"><span style="color: #333399; font-family: Verdana; font-size: 8pt;"> </span></div><div align="center" class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px;"><span style="color: #333399; font-family: Verdana; font-size: 8pt;"> </span></div><div align="center" class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px;"><span style="color: #333399; font-family: Verdana; font-size: 8pt;"><img border="0" height="259" src="http://www.catedu.es/matematicas_mundo/FOTOGRAFIAS/conicas10.jpg" width="400" /></span></div><div align="center" class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px;"><span style="color: #333399; font-family: Verdana; font-size: 8pt;">Generalife. La Alhambra. Granada. </span></div><div align="center" class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px;"><span style="color: #333399; font-family: Verdana; font-size: 8pt;"> </span></div><div align="center" class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px;"><span style="color: #333399; font-family: Verdana; font-size: 8pt;"> </span></div><div align="center" class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px;"><span style="color: #333399; font-family: Verdana; font-size: 8pt;"> <img border="0" height="260" src="http://www.catedu.es/matematicas_mundo/FOTOGRAFIAS/conicas11.jpg" width="400" /></span></div><div align="center" class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px;"><span style="color: #333399; font-family: Verdana; font-size: 8pt;">Puente sobre el Sena. Paris.</span></div><div style="text-align: center;"><br />
</div><div align="center" class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px;"><span style="color: #333399; font-family: Verdana; font-size: 8pt;"><img border="0" height="264" src="http://www.catedu.es/matematicas_mundo/FOTOGRAFIAS/conicas12.jpg" width="400" /></span></div><div align="center" class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px;"><span style="color: #333399; font-family: Verdana; font-size: 8pt;">Murallas romanas y Pza. Lanuza. Zaragoza </span></div><div align="center" class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px;"><br />
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</div><div align="center" class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px;"><img border="0" height="300" src="http://www.catedu.es/matematicas_mundo/FOTOGRAFIAS/conicas27.JPG" width="400" /></div><div align="center" class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px;"><span style="color: #003366; font-family: Verdana; font-size: 8pt;">Fachada de la Iglesia de Santa Isabel. Zaragoza.</span></div><div align="center" class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px;"><span style="color: #333399; font-family: Verdana; font-size: 8pt;"> </span></div><div align="center" class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px;"><br />
</div><div class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px; text-align: center;"><span style="color: #333399; font-family: Verdana; font-size: 8pt;"> <b><i>Círculos y circunferencias </i></b></span><b><i><span style="color: #333399; font-family: Verdana; font-size: 8pt;">(elipses con excentricidad 0) </span></i></b></div><div class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px; text-align: center;"><br />
</div><div class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px; text-align: center;"><img border="0" height="300" src="http://www.catedu.es/matematicas_mundo/FOTOGRAFIAS/conicas29.JPG" width="400" /></div><div class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px; text-align: center;"><span style="color: #333399; font-family: Verdana;"><span style="font-size: 8pt;">Piedra del Sol (Natural History Museum. Nueva York)</span></span></div><div class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px; text-align: center;"><br />
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</div><div class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px; text-align: center;"><span style="font-family: Verdana; font-size: 8pt;"><img border="0" height="300" src="http://www.catedu.es/matematicas_mundo/FOTOGRAFIAS/conicas25.jpg" width="400" /></span></div><div class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px; text-align: center;"><span style="color: #333399; font-family: Verdana; font-size: 8pt;">Palacio Nacional da Pena - Sintra (Portugal)</span></div><div class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px; text-align: center;"><br />
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</div><div class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px; text-align: center;"><span style="font-family: Verdana; font-size: 8pt;"><img border="0" height="299" src="http://www.catedu.es/matematicas_mundo/FOTOGRAFIAS/conicas24.jpg" width="398" /></span></div><div class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px; text-align: center;"><span style="color: navy; font-family: Verdana; font-size: 8pt;"><i>Círculo imperial </i>- Templo del Cielo. Pekín.</span></div><div class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px; text-align: center;"><br />
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</div><div align="center" class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px;"><span style="color: #333399; font-family: Verdana; font-size: 8pt;"><img border="0" height="400" src="http://www.catedu.es/matematicas_mundo/FOTOGRAFIAS/conicas13.jpg" width="272" /> </span></div><div align="center" class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px;"><span style="color: #333399; font-family: Verdana; font-size: 8pt;"> Salón de Embajadores. Reales Alcázares. Sevilla. (*)</span></div><div align="center" class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px;"><span style="color: #333399; font-family: Verdana; font-size: 8pt;"> </span></div><div align="center" class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px;"><span style="color: #333399; font-family: Verdana; font-size: 8pt;"> </span></div><div align="center" class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px;"><span style="color: #333399; font-family: Verdana; font-size: 8pt;"><img border="0" height="264" src="http://www.catedu.es/matematicas_mundo/FOTOGRAFIAS/conicas18.jpg" width="400" /></span></div><div align="center" class="MsoNormal" style="line-height: 100%; margin-bottom: 0px; margin-top: 0px;"><span style="color: #333399; font-family: Verdana; font-size: 8pt;"><i>Círculo protector</i>. Lanzarote. </span></div>Daniel Espinozahttp://www.blogger.com/profile/01853802069444746502noreply@blogger.com1tag:blogger.com,1999:blog-4236088990450264130.post-41607443347495163842010-09-09T08:33:00.000-07:002010-09-09T08:33:04.573-07:00EDISON ROJAS- ARUITECTURE CONICAS<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; margin-right: 1em; text-align: left;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJYmo_sQOmPFzMAev21GeVF2gWgRQxQJx0xpYBaIWq1TNcUyHGmTydk57ALt0Hyh5IIAYXouCrFUtAYg4chXRAQPIYik1ucLnO3UW1z3DLSD3UQj_qBAnzQnyD0nqn-MEIvcxxAxtZb4o/s1600/Conicas2.jpg" imageanchor="1" style="clear: left; cssfloat: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" height="275" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJYmo_sQOmPFzMAev21GeVF2gWgRQxQJx0xpYBaIWq1TNcUyHGmTydk57ALt0Hyh5IIAYXouCrFUtAYg4chXRAQPIYik1ucLnO3UW1z3DLSD3UQj_qBAnzQnyD0nqn-MEIvcxxAxtZb4o/s400/Conicas2.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Añadir leyenda</td></tr>
</tbody></table>En esta entrda explicaremos como las figuras cónicas se presentan en la arquitectura. Con esto nos podemos dar cuenta la importancia de las conicas en la vida cotidianaDaniel Espinozahttp://www.blogger.com/profile/01853802069444746502noreply@blogger.com0tag:blogger.com,1999:blog-4236088990450264130.post-46058675377296596742010-09-09T00:03:00.000-07:002010-09-10T18:51:03.245-07:00Ortega Albujar Eddynson Asterio<a href="http://www.cidse.itcr.ac.cr/cursos-linea/SUPERIOR/t1-conicas/3-Elipse/index.html">http://www.cidse.itcr.ac.cr/cursos-linea/SUPERIOR/t1-conicas/3-Elipse/index.html</a><br />
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<a href="http://www.cidse.itcr.ac.cr/cursos-linea/SUPERIOR/t1-conicas/2-Parabola/index.html">http://www.cidse.itcr.ac.cr/cursos-linea/SUPERIOR/t1-conicas/2-Parabola/index.html</a><br />
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<a href="http://www.cidse.itcr.ac.cr/cursos-linea/SUPERIOR/t1-conicas/4-Hiperbola/index.html">http://www.cidse.itcr.ac.cr/cursos-linea/SUPERIOR/t1-conicas/4-Hiperbola/index.html</a><br />
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<span style="color: red;">ELIPSE</span>:Una elipse es el conjunto de puntos(x,y) cuya suma de distancias a dos puntos distintos prefijados (llamados focos) es constante. <br />
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La recta que pasa por los focos corta a la elipse en dos puntos llamados vértices. La cuerda que une los vértices es el eje mayor de la elipse y su punto medio el centro de la elipse. La cuerda perpendicular al eje mayor y que pasa por el centro se llama eje menor de la elipse.<br />
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Para visualizar la definición de la elipse, basta imaginar dos chinches clavados en los focos y un trozo de cuerda atada a ellos. Al ir moviendo un lápiz que tensa esa cuerda, su trazo irá dibujando una elipse, como se muestra en la figura 1.<br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjp2hXzTfZgtPv007_RDq05CD7a6n-5YJnVchS3UOEstD2lbfMfQoZRBjPn1x1xm8V-WtQ9WeEaH-_k-G05ELB707goUL2QNgGcAnHIq5OdqGtngTT7VhaBYV2OvY-vus9ZctgmO-c4yOw/s1600/Elipse-fig1.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjp2hXzTfZgtPv007_RDq05CD7a6n-5YJnVchS3UOEstD2lbfMfQoZRBjPn1x1xm8V-WtQ9WeEaH-_k-G05ELB707goUL2QNgGcAnHIq5OdqGtngTT7VhaBYV2OvY-vus9ZctgmO-c4yOw/s320/Elipse-fig1.gif" /></a></div><div class="separator" style="clear: both; text-align: center;">figura 1</div><div class="separator" style="clear: both; text-align: center;"><br />
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</div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;">La forma canónica de la ecuación de una elipse de centro (<em>h,k</em>) y ejes mayor y menor de longitudes 2a y 2b respectivamente, con a>b , es</div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"><br />
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</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjeGX1IAbWMmPYRQRwpq4LyvxAvnyEBObl7Nn5bHQY9-1yaDBRAWhbjr54g0tW1EjFI1tYM8rPrJujDVd-VuhIXCANyDmNAT262M4Dz5PyjZOXjsIUJkFeP1BwV5lPKliBsXzpUS1Bheb0/s1600/img8.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjeGX1IAbWMmPYRQRwpq4LyvxAvnyEBObl7Nn5bHQY9-1yaDBRAWhbjr54g0tW1EjFI1tYM8rPrJujDVd-VuhIXCANyDmNAT262M4Dz5PyjZOXjsIUJkFeP1BwV5lPKliBsXzpUS1Bheb0/s320/img8.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
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</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcV5mmwW_9encF0xjM3f2iLexY-KEHP49tDC-JXRR1IJ650OWZGQ1VQVUPk_JQv6FAX9SLRjUNX2JmfjgyhOfO1OFgPga0ekdYLXWzViwqN6Q7MpkqV6JVXDO0smJsrnBmRCMtn0Ds_Hk/s1600/Elipse-fig2.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcV5mmwW_9encF0xjM3f2iLexY-KEHP49tDC-JXRR1IJ650OWZGQ1VQVUPk_JQv6FAX9SLRjUNX2JmfjgyhOfO1OFgPga0ekdYLXWzViwqN6Q7MpkqV6JVXDO0smJsrnBmRCMtn0Ds_Hk/s320/Elipse-fig2.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">Observación : de la figura 2, podemos deducir que d(V1,F1) + d(V1,F2)= 2a, (tomando P=V1)es decir, 2a es la constante a la que se refiere la definición.</div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">Los focos están en el eje mayor a c unidades del centro con <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiL5Rmq5e2mw52FIwLicadNQhymfJw3iHzIyZbzt_N2yFUwZ3ujSoTtlQxZItatV07D8lhE36mbmZ2_nhFXv6lpMjSEfhjey9YPrPJSHY3M_L6GkbQ6KbTn4t2xuwItaDv0sSUfjHOmdeE/s1600/img10.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiL5Rmq5e2mw52FIwLicadNQhymfJw3iHzIyZbzt_N2yFUwZ3ujSoTtlQxZItatV07D8lhE36mbmZ2_nhFXv6lpMjSEfhjey9YPrPJSHY3M_L6GkbQ6KbTn4t2xuwItaDv0sSUfjHOmdeE/s320/img10.gif" /></a>,y el eje mayor es horizontal.En el caso de que el eje mayor sea vertical la ecuación toma la forma: </div><br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhpuICGDRhMzM8wdUv4TyND9NhPluLGETdEQ_xZZrIwvUb0z_bvakE_Y8ikWTspj2LPYbpFbce2WHRX1tds9tQU2yp_EI9-FwvKkY28qVZ8eibVB5CTjABojXTdpBZDwShHo5qKRtDBpQk/s1600/img11.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhpuICGDRhMzM8wdUv4TyND9NhPluLGETdEQ_xZZrIwvUb0z_bvakE_Y8ikWTspj2LPYbpFbce2WHRX1tds9tQU2yp_EI9-FwvKkY28qVZ8eibVB5CTjABojXTdpBZDwShHo5qKRtDBpQk/s320/img11.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">Observación : la demostración de este teorema no es complicada, basta aplicar la definición y la fórmula de distancia (figura 2</div><div class="separator" style="clear: both; text-align: center;"><br />
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</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifaljZqlg0206F6ewlchlsKVc4hUWfPwcUruCpSu_QwlRh7DJFGdeNww8C5hWV7mDXFI7tbTfsHVp6S9jqg8eEIL6jvDMtksbBVOGGJihcWWvkQ8E74_WRPD1Et-VoLL8Kx_N00b27xU8/s1600/img12.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifaljZqlg0206F6ewlchlsKVc4hUWfPwcUruCpSu_QwlRh7DJFGdeNww8C5hWV7mDXFI7tbTfsHVp6S9jqg8eEIL6jvDMtksbBVOGGJihcWWvkQ8E74_WRPD1Et-VoLL8Kx_N00b27xU8/s320/img12.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">Simplificando</div><div class="separator" style="clear: both; text-align: center;"><br />
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</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgfxxlYo3_j9tF4VMpKjgmov-h8gtlOJ_rpXBrGF2yEOrBe0EMbDywl1l-jz4SSASoPF8wyfK1g6NRHMxHUXIWd-ljMd70_7ldCuEseH-cGdpYoPKgtP_pvG-FRhozRjTaJQ_Sxk4LLpYQ/s1600/img13.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgfxxlYo3_j9tF4VMpKjgmov-h8gtlOJ_rpXBrGF2yEOrBe0EMbDywl1l-jz4SSASoPF8wyfK1g6NRHMxHUXIWd-ljMd70_7ldCuEseH-cGdpYoPKgtP_pvG-FRhozRjTaJQ_Sxk4LLpYQ/s320/img13.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">Pero,<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiL5Rmq5e2mw52FIwLicadNQhymfJw3iHzIyZbzt_N2yFUwZ3ujSoTtlQxZItatV07D8lhE36mbmZ2_nhFXv6lpMjSEfhjey9YPrPJSHY3M_L6GkbQ6KbTn4t2xuwItaDv0sSUfjHOmdeE/s1600/img10.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiL5Rmq5e2mw52FIwLicadNQhymfJw3iHzIyZbzt_N2yFUwZ3ujSoTtlQxZItatV07D8lhE36mbmZ2_nhFXv6lpMjSEfhjey9YPrPJSHY3M_L6GkbQ6KbTn4t2xuwItaDv0sSUfjHOmdeE/s320/img10.gif" /></a>y así obtenemos la ecuación canónica de la elipse </div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">Teorema (propiedad de reflexión) </div><br />
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La recta tangente a una elipse en un punto P forma ángulos iguales con las rectas que pasan por P y por alguno de los focos. <br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFmssU5msuZ1srkYEj9FVo1NBKMYllB9XV4dujlb9I-VN3gPJNxF_iDdU1DIV6J40P1x0prgDgorQHXMkmzDnvjE_vWitYyalQ2QJ_GYZShOUKKgFYNCf1HsRVlssE-C-kfkoFQ4LNKvI/s1600/Elipse-fig3.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFmssU5msuZ1srkYEj9FVo1NBKMYllB9XV4dujlb9I-VN3gPJNxF_iDdU1DIV6J40P1x0prgDgorQHXMkmzDnvjE_vWitYyalQ2QJ_GYZShOUKKgFYNCf1HsRVlssE-C-kfkoFQ4LNKvI/s320/Elipse-fig3.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">figura 3</div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">Ejemplo 1 Hallar la ecuación canónica de la elipse </div><div class="separator" style="clear: both; text-align: center;"><br />
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</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUnLqfYdV9VNjgM6NezyVder-R8uZvT-7k_whfJUN45tjPgBEXaIV1ckAceWov48ni7XTBTQqXt8j4_SqGZLqcJCGmcQjFjbmoPcVQaWrPWkeXxbCKn7AHaYa7co11IYo_aaWeHHaZrUQ/s1600/img20.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUnLqfYdV9VNjgM6NezyVder-R8uZvT-7k_whfJUN45tjPgBEXaIV1ckAceWov48ni7XTBTQqXt8j4_SqGZLqcJCGmcQjFjbmoPcVQaWrPWkeXxbCKn7AHaYa7co11IYo_aaWeHHaZrUQ/s320/img20.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">Trazar su gráfica identificando los vértices, los focos, el centro y la excentricidad.</div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">Solución </div><br />
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Para hallar la ecuación canónica debemos completar el cuadrado de la expresión en ambas variables x e y . <br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgfkdWBB0wDK33cmHN5FSesPVmoo4oN73dF8W63b-VMhqLLTAPqLZrveuBnVDCdrdg9QSNi7Lk3gWxONsEuzGpUa-MdcsF-n7flZwWz2UVKZeHir5oJVGADOfANg-pjetTskR93Up6K-GA/s1600/img23.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgfkdWBB0wDK33cmHN5FSesPVmoo4oN73dF8W63b-VMhqLLTAPqLZrveuBnVDCdrdg9QSNi7Lk3gWxONsEuzGpUa-MdcsF-n7flZwWz2UVKZeHir5oJVGADOfANg-pjetTskR93Up6K-GA/s320/img23.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">De donde obtenemos que el centro es (1,-2), el valor de a=4 ( es la longitud mayor, esto nos dice que la elipse es vertical), el valor de b=2 y el valor de está dado por : </div><div class="separator" style="clear: both; text-align: center;"><br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhF2gMQkJzLipNbLUSRj4wxLQBM3RzvMVoW6o0GHkZ62j7ismswDpIAZ8p-jh-aPGqh8cROgvMv6N7ktqbT8-Q0EDflQHT3qpHtdF_2gHZxIgJfAM0EsT4DQpTv39sUnIRy1cTSz76WZao/s1600/img28.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhF2gMQkJzLipNbLUSRj4wxLQBM3RzvMVoW6o0GHkZ62j7ismswDpIAZ8p-jh-aPGqh8cROgvMv6N7ktqbT8-Q0EDflQHT3qpHtdF_2gHZxIgJfAM0EsT4DQpTv39sUnIRy1cTSz76WZao/s320/img28.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">Y así, los focos están dados por <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi94zUouWwzldjY0XjKdcZG63sdsWKpWGIjKkZPUCY-tnNl5PHqz3RUdQT5jjGDaE_U1EIpmPu1Sz_XT2Z7ulmzEfeD31la5ut7ja1KxqwYb6_oB0m6uEwKrjXAC1n6_UbROv9shrtZsCc/s1600/img29.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi94zUouWwzldjY0XjKdcZG63sdsWKpWGIjKkZPUCY-tnNl5PHqz3RUdQT5jjGDaE_U1EIpmPu1Sz_XT2Z7ulmzEfeD31la5ut7ja1KxqwYb6_oB0m6uEwKrjXAC1n6_UbROv9shrtZsCc/s320/img29.gif" /></a>y los vértices por <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhrxIK3PZb2e0K5YL0m4dXFmQaGzfM26mPlx7MmYrWyX5kMGYYJUWAbEb3V2OYZc_t2efrvfcdPp-lxt_hi53Mw4CfqhFGPQBabSKLBCm_5HBKbJyCmAOifK16VEFmHsbD59CINIb4JPXg/s1600/img30.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhrxIK3PZb2e0K5YL0m4dXFmQaGzfM26mPlx7MmYrWyX5kMGYYJUWAbEb3V2OYZc_t2efrvfcdPp-lxt_hi53Mw4CfqhFGPQBabSKLBCm_5HBKbJyCmAOifK16VEFmHsbD59CINIb4JPXg/s320/img30.gif" /></a>Por último, la excentricidad es </div><div class="separator" style="clear: both; text-align: center;"><br />
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</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-eVk0gTCr34v2gxZApZOnO7NQZBY-4lWjVjRLuVk9IUCQpruUjX2WfGHW-LsDgHFWLWySTsklsH2GubSWGEU6zxxNRE5Cv9NC7G1KZR9HFEfV2F46KFK77wDBkgAjm3qJt-7Seky1gmc/s1600/img31.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-eVk0gTCr34v2gxZApZOnO7NQZBY-4lWjVjRLuVk9IUCQpruUjX2WfGHW-LsDgHFWLWySTsklsH2GubSWGEU6zxxNRE5Cv9NC7G1KZR9HFEfV2F46KFK77wDBkgAjm3qJt-7Seky1gmc/s320/img31.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">La gráfica se muestra en la figura 4. </div><div class="separator" style="clear: both; text-align: center;"><br />
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</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiq8hM4oa8WZ2NFtlzPeI22HReYyoBG9rISCs2fI54u3-y1zjRMVen3VWwYycQ_FuPm6E7z9X80hQFofzdzBIrRCCbbYkZTj7fZhNJtyxPuxTpqBvh7jL_9_6tdhNdNK_4DqNFoV0Kn80M/s1600/Elipse-fig4.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiq8hM4oa8WZ2NFtlzPeI22HReYyoBG9rISCs2fI54u3-y1zjRMVen3VWwYycQ_FuPm6E7z9X80hQFofzdzBIrRCCbbYkZTj7fZhNJtyxPuxTpqBvh7jL_9_6tdhNdNK_4DqNFoV0Kn80M/s320/Elipse-fig4.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">Definición (excentricidad) </div><br />
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<div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;">La excentricidad e de una elipse está dada por el cociente</div><br />
<div class="separator" style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none; clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6o5oOtisSvlAD-aB-9hemmlAI2__IAmffJTb_wLDZMADBuIVgduoq93pxO8bOi583BxkaqznKvR7EnLsmHDHtQm-g_6eb1nosl4_tq5ht8suzSznjxBqYDnZrYv9okx-m8-kb92l8AWU/s1600/img15.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6o5oOtisSvlAD-aB-9hemmlAI2__IAmffJTb_wLDZMADBuIVgduoq93pxO8bOi583BxkaqznKvR7EnLsmHDHtQm-g_6eb1nosl4_tq5ht8suzSznjxBqYDnZrYv9okx-m8-kb92l8AWU/s320/img15.gif" /></a></div><div class="separator" style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none; clear: both; text-align: center;"><br />
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</div><div class="separator" style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none; clear: both; text-align: center;"><span style="color: red;">Parábola </span></div><br />
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Ahora, vamos a deducir las ecuaciones de las secciones cónicas a partir de su definición como lugares geométricos y no como la intersección de un cono con un plano, como se hizo en la antigüedad. Ya conocemos que la gráfica de una función cuadrática<br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhxfs85pPq97R9fqoIvjwcQ1QPHFYoEIBo1GEBC0fB6-nyaaZQT3JhytaHdrGpQjA0yA9sBPPdZjOOxScVR2AQzpmOpU3o7YNRvVc24yIhNArjaNTkGCrH5E3bC1Pr4-ogV-fzHdxc2V4Y/s1600/img1.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhxfs85pPq97R9fqoIvjwcQ1QPHFYoEIBo1GEBC0fB6-nyaaZQT3JhytaHdrGpQjA0yA9sBPPdZjOOxScVR2AQzpmOpU3o7YNRvVc24yIhNArjaNTkGCrH5E3bC1Pr4-ogV-fzHdxc2V4Y/s320/img1.gif" /></a></div><br />
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con <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjStWZYxfRPQY0G8-Hcbxpi3a0-WJYqY2t_GXEEIOn85sefjGkcjdF4xH3dGQT3D1TTquacFPM6koadkw45vOHbvBeUcRI3wTczN16uQ-4qjvUdLjERPdWBY-2Lgcx5rvMRsyspdRFRsrA/s1600/img2.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjStWZYxfRPQY0G8-Hcbxpi3a0-WJYqY2t_GXEEIOn85sefjGkcjdF4xH3dGQT3D1TTquacFPM6koadkw45vOHbvBeUcRI3wTczN16uQ-4qjvUdLjERPdWBY-2Lgcx5rvMRsyspdRFRsrA/s320/img2.gif" /></a> , es una parábola. Sin embargo, no toda parábola es la gráfica de una función, como podemos concluir de la siguiente definición. <br />
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Definición <br />
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</div>Una parábola es el conjunto de puntos P(x,y) en el plano que equidistan de un punto fijo F (llamado foco de la parábola) y de una recta fija L (llamada la directriz de la parábola) que no contiene a F (figura 1). <br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYlKMQkB9la0LWOY476k-9g998RVcTNRi8NG96xxQNX-bDtsH7rO_9K-GObLh6idt1e7XeWD-c9Fx1I7ykQLoyZ3X_emGygbOM9ASoTXl336sdvUT5tdRy49QgQP4arMcXLQ0yNg2Y0tQ/s1600/parabola-fig1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYlKMQkB9la0LWOY476k-9g998RVcTNRi8NG96xxQNX-bDtsH7rO_9K-GObLh6idt1e7XeWD-c9Fx1I7ykQLoyZ3X_emGygbOM9ASoTXl336sdvUT5tdRy49QgQP4arMcXLQ0yNg2Y0tQ/s320/parabola-fig1.jpg" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">El punto medio entre el foco y la directriz se llama vértice, la recta que pasa por el foco y por el vértice se llama eje de la parábola.Se puede observar en la figura 1 que una parábola es simétrica respecto a su eje. </div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">Teorema (ecuación canónica de la parábola) </div><br />
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<div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHw2S4t88SODdcM51u_fPPvesJAzmwRFYg9unLqXyN1H64HFQV2Yk28Ief6Eq66s32hAbytUnicLVRVkiaY1gnbvsmaRxzY33XyXURDfuzNHdWmgJ8o1nayNnwoLYYnObM-4k4C1yBrTY/s1600/img7.gif" imageanchor="1" style="clear: right; cssfloat: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHw2S4t88SODdcM51u_fPPvesJAzmwRFYg9unLqXyN1H64HFQV2Yk28Ief6Eq66s32hAbytUnicLVRVkiaY1gnbvsmaRxzY33XyXURDfuzNHdWmgJ8o1nayNnwoLYYnObM-4k4C1yBrTY/s320/img7.gif" /></a>La forma canónica de la ecuación de una parábola con vértice <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjHmT11bhIgM1YfATxxmhwJwS1357pqxjYDtiDx9U6urgUUEJIS-mZ8Vinr_QMkVbhrnPsr4w6uLxQFEHBB0nSqjrBrsVt1xb_xm5Phrs-Ro7IYTDLwaOD1WvELQyRBmOmOvtsoP2R1MrQ/s1600/img6.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjHmT11bhIgM1YfATxxmhwJwS1357pqxjYDtiDx9U6urgUUEJIS-mZ8Vinr_QMkVbhrnPsr4w6uLxQFEHBB0nSqjrBrsVt1xb_xm5Phrs-Ro7IYTDLwaOD1WvELQyRBmOmOvtsoP2R1MrQ/s320/img6.gif" /></a>y directriz </div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"> es:</div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"><br />
</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi8of9dvyJBpVIssGxMmfB96eXYkWM7-dN9rLmh6SFw-v1XhUK2LuW-6TiU5nqyEhO8KPwTwIhTgO03vb0IsdRY_P9OnN_cWZzKH5aAP2AM7xolOnpIXMvOX6CxYCdzMIA8BIvnUAqQLOc/s1600/img8.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi8of9dvyJBpVIssGxMmfB96eXYkWM7-dN9rLmh6SFw-v1XhUK2LuW-6TiU5nqyEhO8KPwTwIhTgO03vb0IsdRY_P9OnN_cWZzKH5aAP2AM7xolOnpIXMvOX6CxYCdzMIA8BIvnUAqQLOc/s320/img8.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">El eje de la parábola es vertical y el foco F está a <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOH2jfh5zaSvAu6mkUCnwf9X1wkwdJGAqU4oKjwDOcyIuOVM3zQNJ0a8XL7xbeY9cpfFTkRHTkYF22yN-w9k5iPnN6haI9Sj59vsPqr4NEEhxof9iSlhf6Dz9dbTpknqe0p54eagDRAlk/s1600/img9.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOH2jfh5zaSvAu6mkUCnwf9X1wkwdJGAqU4oKjwDOcyIuOVM3zQNJ0a8XL7xbeY9cpfFTkRHTkYF22yN-w9k5iPnN6haI9Sj59vsPqr4NEEhxof9iSlhf6Dz9dbTpknqe0p54eagDRAlk/s320/img9.gif" /></a>unidades (orientadas) del vértice. Si p>0 , la parábola abre hacia arriba y el foco está en <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhx3eLDIkz05oF63Ss3vz1gGPaKwOapiRp27hhS4VK8KUKNPYZvO4n6wIS2CnA9G0gB-JiGM7Ytoo_TPoazxKh_ZM-pgSxQgFfEXfXW2shFeTJp-_CFPkf-rZJ9yjBhaeIW6Jv3VXVPmW4/s1600/img11.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhx3eLDIkz05oF63Ss3vz1gGPaKwOapiRp27hhS4VK8KUKNPYZvO4n6wIS2CnA9G0gB-JiGM7Ytoo_TPoazxKh_ZM-pgSxQgFfEXfXW2shFeTJp-_CFPkf-rZJ9yjBhaeIW6Jv3VXVPmW4/s320/img11.gif" /></a>; si p<0 , la parábola abre hacia abajo y el foco está en <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhY-tyBnXgMcBKMHa5XcBSJ6StC2MGMSTDgRL6QvVelGWBmFSUlbbISnut3-7lUkpBTlHSjqfWQl5azJ9txkE0I2l3pfEtRhQF0NPdgQb6QABAqBsJjSsZRCjmElBRzf5yNAnimHH0Gkn4/s1600/img13.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhY-tyBnXgMcBKMHa5XcBSJ6StC2MGMSTDgRL6QvVelGWBmFSUlbbISnut3-7lUkpBTlHSjqfWQl5azJ9txkE0I2l3pfEtRhQF0NPdgQb6QABAqBsJjSsZRCjmElBRzf5yNAnimHH0Gkn4/s320/img13.gif" /></a>Si la directriz es <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEivfNeJ5XFqhlkkeLjilva9PL2rDd-VgvJa9fXrFyiYMlW_KXSIczS8aG7YfuSLBRL-6yXNir189PfMLuIOYynf8HG9lKAVH6xFoz2TYyMBx0JimrT98Xmrr7N_T91SUkqgbZ6mLGKaVeA/s1600/img14.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEivfNeJ5XFqhlkkeLjilva9PL2rDd-VgvJa9fXrFyiYMlW_KXSIczS8aG7YfuSLBRL-6yXNir189PfMLuIOYynf8HG9lKAVH6xFoz2TYyMBx0JimrT98Xmrr7N_T91SUkqgbZ6mLGKaVeA/s320/img14.gif" /></a>(eje horizontal), la ecuación es </div><div class="separator" style="clear: both; text-align: center;"><br />
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</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgz8fE2pL1L7Jy-MVrRMmqynGs_f8u_qEW8hMY5SW_fyu4txCIiQH1YH5SGfaBuaWaiIRvCXAR9Xd_ErO5jwFmm5IPaGlSs4kuuQpkfKXFzV4vaNckT3gP6ZJKMZ9I4ugb9MeAQnbuj-mc/s1600/img15.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgz8fE2pL1L7Jy-MVrRMmqynGs_f8u_qEW8hMY5SW_fyu4txCIiQH1YH5SGfaBuaWaiIRvCXAR9Xd_ErO5jwFmm5IPaGlSs4kuuQpkfKXFzV4vaNckT3gP6ZJKMZ9I4ugb9MeAQnbuj-mc/s320/img15.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">El eje de la parábola es horizontal y el foco F está a <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOH2jfh5zaSvAu6mkUCnwf9X1wkwdJGAqU4oKjwDOcyIuOVM3zQNJ0a8XL7xbeY9cpfFTkRHTkYF22yN-w9k5iPnN6haI9Sj59vsPqr4NEEhxof9iSlhf6Dz9dbTpknqe0p54eagDRAlk/s1600/img9.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOH2jfh5zaSvAu6mkUCnwf9X1wkwdJGAqU4oKjwDOcyIuOVM3zQNJ0a8XL7xbeY9cpfFTkRHTkYF22yN-w9k5iPnN6haI9Sj59vsPqr4NEEhxof9iSlhf6Dz9dbTpknqe0p54eagDRAlk/s320/img9.gif" /></a>unidades (orientadas) del vértice. Si p>0 , la parábola abre hacia la derecha y el foco está en <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvNIljDpHx4STPDYgzWwQ9_3mpUSuNzuUEX2TooCHiOVUnfDjnVJyCknA_J7FxQ91xpqLCMEQWCYXvVTJo2uKz-qb47XbkihNb8tkibfPRPtSpDqLO4ianXBgatVgNTGira9F_13L5EN8/s1600/img16.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvNIljDpHx4STPDYgzWwQ9_3mpUSuNzuUEX2TooCHiOVUnfDjnVJyCknA_J7FxQ91xpqLCMEQWCYXvVTJo2uKz-qb47XbkihNb8tkibfPRPtSpDqLO4ianXBgatVgNTGira9F_13L5EN8/s320/img16.gif" /></a>; si p<0 parábola abre hacia la izquierda y el foco está en <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1pvmsg7GQEqt51liN1ULRWJYLPx0qkdDzSaPZMrCkIyMRjarNNBcwz_3bjtkQsl1LGmsiJ_eBVUfMHwr8OesMr4aNwETAIAJmLZX524T4-qcoTopIl1e0dIphRwgqLQ6kV3SJev95tl0/s1600/img17.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1pvmsg7GQEqt51liN1ULRWJYLPx0qkdDzSaPZMrCkIyMRjarNNBcwz_3bjtkQsl1LGmsiJ_eBVUfMHwr8OesMr4aNwETAIAJmLZX524T4-qcoTopIl1e0dIphRwgqLQ6kV3SJev95tl0/s320/img17.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">Observación : la demostración de este teorema no es difícil, basta aplicar la definición y la fórmula de distancia (figura 1).Para el caso en el cual el eje de la parábola es vertical, tenemos que </div><div class="separator" style="clear: both; text-align: center;"><br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi36nFCo2QTOPt4qn3nc-CelSI9aniY6HI7b9RE-gkSIlz7Uk3Qh9a9zssgwtlju1XBN2G4NpQD6XAL_xI9UgaNQY3KYlIYSPOqpvq1hs_46FJfLajsNeDhMxLjFNmowF1nlTzkG-SSDPE/s1600/img18.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi36nFCo2QTOPt4qn3nc-CelSI9aniY6HI7b9RE-gkSIlz7Uk3Qh9a9zssgwtlju1XBN2G4NpQD6XAL_xI9UgaNQY3KYlIYSPOqpvq1hs_46FJfLajsNeDhMxLjFNmowF1nlTzkG-SSDPE/s320/img18.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">Ejemplo 1. </div><br />
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Trazar la gráfica y hallar la ecuación canónica, el vértice, el foco y la directriz de la parábola cuya ecuación es <br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhiNvLNJpWXWzXNlJjcmDW_aJERM-Nii9oEAoQQAdaImhqy9vEny2tDIvngpyuHYJVXzxzrZumKVWPa6ClohclHVtnPAmGQB2Gg2MKcCX6M2xYApPwtE8ytCaJlkjuX2BU8XIYjM45g57s/s1600/img19.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhiNvLNJpWXWzXNlJjcmDW_aJERM-Nii9oEAoQQAdaImhqy9vEny2tDIvngpyuHYJVXzxzrZumKVWPa6ClohclHVtnPAmGQB2Gg2MKcCX6M2xYApPwtE8ytCaJlkjuX2BU8XIYjM45g57s/s320/img19.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">Solución</div><br />
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Para hallar la ecuación canónica debemos completar el cuadrado en a. De la ecuación de la parábola tenemos que <br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi7NfB763-RsLnBqqBw0rcjWNDdnaVKUlKGPf9KEVEUEy9q2lXNsLyhqbhiKuN15ETC2dxx75NOxff8zbPBCUWErDC9e-ds5tv0MtYG6QeFT5oaZIdRlML_OlL1SqQcSOCUohSsi-TtRNE/s1600/img20.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi7NfB763-RsLnBqqBw0rcjWNDdnaVKUlKGPf9KEVEUEy9q2lXNsLyhqbhiKuN15ETC2dxx75NOxff8zbPBCUWErDC9e-ds5tv0MtYG6QeFT5oaZIdRlML_OlL1SqQcSOCUohSsi-TtRNE/s320/img20.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">De donde obtenemos que p=1 y el vértice v=(2,3) , por lo tanto, la parábola abre hacia la derecha y tiene el foco en F=(3,3) , la recta directriz es x=1 . La gráfica se muestra en la figura 2.</div><div class="separator" style="clear: both; text-align: center;"><br />
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</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJMdeur_-_9BkO1aX0W4LMwDuuoKJuNXiDmccrrQwaFAwNUxmL5XIs41KJCQvp8xpR0BGhkcgA2K0vMpMJ0scu6MXY91Ot9K_t8rB51eGYh_MjSCl9gm4NQ-JNA33XlTlesjG4HOer2vc/s1600/parabola-fig2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJMdeur_-_9BkO1aX0W4LMwDuuoKJuNXiDmccrrQwaFAwNUxmL5XIs41KJCQvp8xpR0BGhkcgA2K0vMpMJ0scu6MXY91Ot9K_t8rB51eGYh_MjSCl9gm4NQ-JNA33XlTlesjG4HOer2vc/s320/parabola-fig2.jpg" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
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</div><div class="separator" style="clear: both; text-align: center;">Hipérbola</div><br />
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Las hipérbolas aparecen en muchas situaciones reales, por ejemplo, un avión que vuela a velocidad supersónica paralelamente a la superficie de la tierra, deja una huella acústica hiperbólica sobre la superficie. La intersección de una pared y el cono de luz que emana de una lámpara de mesa con pantalla troncocónica, es una hipérbola.<br />
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La definición de la hipérbola como lugar geométrico es similar a la dada para la elipse, como vemos en seguida<br />
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Definición <br />
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Una hipérbola es el conjunto de puntos P(x,y) para los que la diferencia de sus distancias a dos puntos distintos prefijados (llamados focos) es constante. <br />
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La recta que pasa por los focos corta a la hipérbola en dos puntos llamados vértices. El segmento recto que une los vértices se llama eje transversal y su punto medio es el centro de la hipérbola. Un hecho distintivo de la hipérbola es que su gráfica tiene dos partes separadas, llamadas ramas.<br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHn0NZD6el73gmvf6P74har8bnSlAOAMEHAKG2TemW6uQs5hAU2HSXIKjHePdwlwqouZq1g_6H79zUWK_9_fWsrrDK5ajH2TfJMkZE4IYy2UJIznzdMAMnGxvB8DvkcYuq45W0FHPv_5U/s1600/hiperbola-fig0.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHn0NZD6el73gmvf6P74har8bnSlAOAMEHAKG2TemW6uQs5hAU2HSXIKjHePdwlwqouZq1g_6H79zUWK_9_fWsrrDK5ajH2TfJMkZE4IYy2UJIznzdMAMnGxvB8DvkcYuq45W0FHPv_5U/s320/hiperbola-fig0.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">Teorema (ecuación canónica de la hipérbola) </div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"></div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"><br />
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</div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none; clear: both; text-align: center;"> La ecuación canónica de la hipérbola con centro en (<em>h,k</em>) es</div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none; clear: both; text-align: center;"><br />
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</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9q9tJbnEC1sno3n2zbhDNqQSBHOVNrOZgCMKNixb9CFN9KlmqDjnuUC7Yq63PWod_5HtBFoHAyY7TmumHVtsMw6nc5pGVOTrWHFECbJN7wI-KtgNyMFShukSvLYlSgt1O5eOeMFAe2NM/s1600/img3.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9q9tJbnEC1sno3n2zbhDNqQSBHOVNrOZgCMKNixb9CFN9KlmqDjnuUC7Yq63PWod_5HtBFoHAyY7TmumHVtsMw6nc5pGVOTrWHFECbJN7wI-KtgNyMFShukSvLYlSgt1O5eOeMFAe2NM/s320/img3.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">con eje transversal horizontal. Y </div><br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcl1GXlxBEOrvO-990Zsi2ryVn-3Jp07FdZaqzpIDWq1F-sGnjWR_dC8n4tE3p1LaYDpfieTUFJp85t1dI-L68ItMKKeIV548f4ax5f8yD0yANgnue-Q0DQU4ORzAUEON-DoWcAjtgxH0/s1600/img4.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcl1GXlxBEOrvO-990Zsi2ryVn-3Jp07FdZaqzpIDWq1F-sGnjWR_dC8n4tE3p1LaYDpfieTUFJp85t1dI-L68ItMKKeIV548f4ax5f8yD0yANgnue-Q0DQU4ORzAUEON-DoWcAjtgxH0/s320/img4.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">con eje transversal vertical. </div><br />
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Los vértices están a una distancia de a unidades del centro y los focos a una distancia de c unidades del centro. Además <br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnrwvqcH3dTLD0eUQqsl_GbyxTXqYWsms4C0bc7dTGJu09KeHHhnKkEqxqtfMEvqfYokBC_K0o9UlMb_fHxWbOUwWJeghVY-4MV1Y4Ctg23C4ejbgDddujCzHDPhnQzVdocUEu5z3Q1sM/s1600/img5.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnrwvqcH3dTLD0eUQqsl_GbyxTXqYWsms4C0bc7dTGJu09KeHHhnKkEqxqtfMEvqfYokBC_K0o9UlMb_fHxWbOUwWJeghVY-4MV1Y4Ctg23C4ejbgDddujCzHDPhnQzVdocUEu5z3Q1sM/s320/img5.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
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</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPvDC10vFpz9-0lmB4RgIbPvrceGgJkLQQRlkQLZveEX8XbhhTot8IIPxJvzLU_s5kBV6sYKX4ycVnf6oEGKF8Cw1teXxA72MbZbSmJsiR44Qawvt5oDYAg6DKWZ4WZrGsjwFM1wVorlY/s1600/hiperbola-fig1.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPvDC10vFpz9-0lmB4RgIbPvrceGgJkLQQRlkQLZveEX8XbhhTot8IIPxJvzLU_s5kBV6sYKX4ycVnf6oEGKF8Cw1teXxA72MbZbSmJsiR44Qawvt5oDYAg6DKWZ4WZrGsjwFM1wVorlY/s320/hiperbola-fig1.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">Resumiendo:</div><br />
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Si el eje transversal de la hipérbola es horizontal entonces <br />
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El centro está en <em>(h,k)</em> <br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjN9ACDetTcnSmu9b5WmTtAJhX_0VmPhC2cc9wPMZAw6FuSoXf_W10Pq7eN1NgjabH3OLiP_ynx19O2_YIlRjN1bG3u4wGRcpIxybEsQoqPmF_u41s75R-IVgvQx85Bj10sfdQWLrZHzlw/s1600/img7.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjN9ACDetTcnSmu9b5WmTtAJhX_0VmPhC2cc9wPMZAw6FuSoXf_W10Pq7eN1NgjabH3OLiP_ynx19O2_YIlRjN1bG3u4wGRcpIxybEsQoqPmF_u41s75R-IVgvQx85Bj10sfdQWLrZHzlw/s320/img7.gif" /></a> </div>Los vértices están en <br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNeD_xpZe_1fIJuTrxBJlq2VWHurlDgH35Yk-y91CMQ2Gf6PG8cwNtYCjoO1GaHcLiGM22wV8k0obO0B6xIzRIiA9771A2-telDCaN-PCpml5S4b8Xi78ImqyjgOm5xqNINdhVl8rsmlM/s1600/img8.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNeD_xpZe_1fIJuTrxBJlq2VWHurlDgH35Yk-y91CMQ2Gf6PG8cwNtYCjoO1GaHcLiGM22wV8k0obO0B6xIzRIiA9771A2-telDCaN-PCpml5S4b8Xi78ImqyjgOm5xqNINdhVl8rsmlM/s320/img8.gif" /></a></div>Los focos están en <br />
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Si el eje transversal de la hipérbola es vertical entonces <br />
El centro está en<em> (h,k) </em><br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijekVAT_QgjpBbM8pvbS0CuFapsMxbaEhTj5nILFzatMUecXx-7_ayWcWhNm5TvxvXHQsF82Np7FHD5W_jS4bxGRZSi6W2l5PczUxb2On3DktKXjDud2p78RR-vD1mXdkkKTMlLYKxDFU/s1600/img9.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijekVAT_QgjpBbM8pvbS0CuFapsMxbaEhTj5nILFzatMUecXx-7_ayWcWhNm5TvxvXHQsF82Np7FHD5W_jS4bxGRZSi6W2l5PczUxb2On3DktKXjDud2p78RR-vD1mXdkkKTMlLYKxDFU/s320/img9.gif" /></a> </div>Los vértices están en<br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhgKck9E-8-Go_VDVikPwldgrUGVWsMt6Ht4CWuhvWa-_7VMTG8az8-VEudHhFvHiErWB2HWtkJmakpJMGGYVw5r6VGtKfmfS2xCVkuEAJfHu7ZigGV52UnrcBHz59hzVOEiLxht70SHeA/s1600/img10.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhgKck9E-8-Go_VDVikPwldgrUGVWsMt6Ht4CWuhvWa-_7VMTG8az8-VEudHhFvHiErWB2HWtkJmakpJMGGYVw5r6VGtKfmfS2xCVkuEAJfHu7ZigGV52UnrcBHz59hzVOEiLxht70SHeA/s320/img10.gif" /></a> </div>Los focos están en <br />
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Una ayuda importante para trazar la gráfica de una hipérbola son sus asíntotas. Toda hipérbola tiene dos asíntotas que se intersecan en su centro y pasan por los vértices de un rectángulo de dimensiones 2a y 2b y centro en<em> (h,k)</em> .El segmento recto de longitud 2b que une <br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiolQQZz4pF38eyhJzrVi5kuxEdrwpfbOAitA3r9dsE0VwpC9tXlBX3Ws5YTd2-3Ki-Iz919tiobKxgfXXK9cH_bbDjlaIygjKCHkpSNlbh48PuaE0GvaF3UbneOIx_sAAkHb47CwTu6Q0/s1600/img11.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiolQQZz4pF38eyhJzrVi5kuxEdrwpfbOAitA3r9dsE0VwpC9tXlBX3Ws5YTd2-3Ki-Iz919tiobKxgfXXK9cH_bbDjlaIygjKCHkpSNlbh48PuaE0GvaF3UbneOIx_sAAkHb47CwTu6Q0/s320/img11.gif" /></a> se llama eje conjugado de la hipérbola. El siguiente teorema identifica la ecuación de las asíntotas. </div><div class="separator" style="clear: both; text-align: center;"><br />
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</div><div class="separator" style="clear: both; text-align: center;">Teorema (Asíntotas de una hipérbola) </div><br />
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Si la hipérbola tiene un eje transversal horizontal, las ecuaciones de las asíntotas son <br />
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</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguOfSidvZprnQbbDraDkVHDdrC06Do552yzBFCCZKPdxEfsa4rDquscbm_a1MDtLvt9P47tv9hmXOZWK-AlBN4WyeiAuioEnmuJqsBUYu7ZewE_5fP_mLfYdG4i3720aIX8iRtLmyHs_o/s1600/img12.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguOfSidvZprnQbbDraDkVHDdrC06Do552yzBFCCZKPdxEfsa4rDquscbm_a1MDtLvt9P47tv9hmXOZWK-AlBN4WyeiAuioEnmuJqsBUYu7ZewE_5fP_mLfYdG4i3720aIX8iRtLmyHs_o/s320/img12.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">y si el eje transversal es vertical, las ecuaciones de las asíntotas son </div><div class="separator" style="clear: both; text-align: center;"><br />
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</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiECeaiTvqjLAxsWFuaFwoI5wSCDV5TRe0qmgOhB0x0MFj4aJfnijsRpOWmY1FqD2RfWO5Zh0t7sMDX0BrqBiSx_oHJsB5ueZQvyQFgMPVfIsJzg03Gv2neRGWBAaVZrIhpOXL7Nz08OJs/s1600/img13.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiECeaiTvqjLAxsWFuaFwoI5wSCDV5TRe0qmgOhB0x0MFj4aJfnijsRpOWmY1FqD2RfWO5Zh0t7sMDX0BrqBiSx_oHJsB5ueZQvyQFgMPVfIsJzg03Gv2neRGWBAaVZrIhpOXL7Nz08OJs/s320/img13.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">Observación : las asíntotas de la hipérbola coinciden con las diagonales del rectángulo de dimensiones 2a y 2b centro<em> (h,k)</em> Esto sugiere una forma simple de trazar tales asíntotas</div><div class="separator" style="clear: both; text-align: center;"><br />
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</div><div class="separator" style="clear: both; text-align: center;">Definición (excentricidad de una hipérbola) </div><br />
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<div class="separator" style="clear: both; text-align: center;"> <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgf6UVPFtRSwinxF-BOftNaHXQI8xC9yrxqUhPw0Rj0Aae1jwJYI5AFsWSDZiFi5VIxb_CrUZofqw1ExKYfCg5Nr-am01c_lrK614d6y-72rRC6u_7ZtrqHVwWQYeGdje29sMaIZPdaV1I/s1600/img17.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgf6UVPFtRSwinxF-BOftNaHXQI8xC9yrxqUhPw0Rj0Aae1jwJYI5AFsWSDZiFi5VIxb_CrUZofqw1ExKYfCg5Nr-am01c_lrK614d6y-72rRC6u_7ZtrqHVwWQYeGdje29sMaIZPdaV1I/s320/img17.gif" /></a> </div>La excentricidad de una hipérbola está dada por el cociente <br />
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Si la excentricidad es grande los focos están cerca del centro y las ramas de la hipérbola son casi rectas verticales. Si la excentricidad es cercana a uno los focos están lejos del centro y la ramas de la hipérbola son más puntiagudas. <br />
La propiedad reflectora de la hipérbola afirma que un rayo de luz dirigido a uno de los focos de una hipérbola se refleja hacia el otro foco (figura 2). <br />
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Teorema (propiedad de reflexión) <br />
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La tangente en un punto P de una hipérbola es la bisectriz del ángulo formado por lo segmentos que unen este punto con los focos. <br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMF5TxDiSKHAkwd5Yo75kjJyrFIwv9ZqauMobsdK9WlWnaM_sZjnORbvD_E-4B-WBRyCKJiG-GK401vhwUAyq_0wa5rTkO99nbFD84FQRTUqGFE5deTwTySQsQ_U5chtR8KoyKo78ZK6E/s1600/hiperbola-fig2.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMF5TxDiSKHAkwd5Yo75kjJyrFIwv9ZqauMobsdK9WlWnaM_sZjnORbvD_E-4B-WBRyCKJiG-GK401vhwUAyq_0wa5rTkO99nbFD84FQRTUqGFE5deTwTySQsQ_U5chtR8KoyKo78ZK6E/s320/hiperbola-fig2.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
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</div><div class="separator" style="clear: both; text-align: center;">Ejemplo 1</div><br />
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Hallar la ecuación canónica, los focos, los vértices, la excentricidad y las asíntotas de la hipérbola cuya ecuación es <br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhX1lJZ5ihAoJnlst68Kv06fUpcdy36ZRwMzgOrUQhq42jmNS0RuMnQJD9XDKOaH968z3ApVuAmawMzJdJQ1-f3PxEc0TUwUjSVi02FdIfIiPrHlnSLgDpblyC-YJJahkQNJJpLbUxvlgY/s1600/img18.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhX1lJZ5ihAoJnlst68Kv06fUpcdy36ZRwMzgOrUQhq42jmNS0RuMnQJD9XDKOaH968z3ApVuAmawMzJdJQ1-f3PxEc0TUwUjSVi02FdIfIiPrHlnSLgDpblyC-YJJahkQNJJpLbUxvlgY/s320/img18.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">Solución</div><br />
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Completando el cuadrado en ambas variables<br />
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<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhP8wc9JOvUrKJ6-qnNecqGpuIAE9dpvbL4_36WsTKpJ5ZGz97SCQwkknCiwTo-IIBLvB46SdDY54Ssbbr7Rk7H3blpkA6y7fYC6uDCpSvmt0h3e0nM5KMfuRpr0P2EisfBiIjL3vX4AhU/s1600/img19.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhP8wc9JOvUrKJ6-qnNecqGpuIAE9dpvbL4_36WsTKpJ5ZGz97SCQwkknCiwTo-IIBLvB46SdDY54Ssbbr7Rk7H3blpkA6y7fYC6uDCpSvmt0h3e0nM5KMfuRpr0P2EisfBiIjL3vX4AhU/s320/img19.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">Por tanto, el centro está en (2,-3) El eje de la hipérbola es horizontal a=1, b=3 y </div><div class="separator" style="clear: both; text-align: center;"><br />
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</div><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgwAYQIZr8KAG0-_NIkwv24yGRRUsiv4pp73_tBSXqH_86n9hHNN_cCFSXubFPIFsc3ZSW3yZXPB7Un1Ss-UmObm9lUi3dJFgUCXLXBnxd9_kKkNT5muBsgQblev0cGP5JF6evu6lMJ-Eg/s1600/img22.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgwAYQIZr8KAG0-_NIkwv24yGRRUsiv4pp73_tBSXqH_86n9hHNN_cCFSXubFPIFsc3ZSW3yZXPB7Un1Ss-UmObm9lUi3dJFgUCXLXBnxd9_kKkNT5muBsgQblev0cGP5JF6evu6lMJ-Eg/s320/img22.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;">Los vértices están en (1,-3), (3,-3) los focos en <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjW1eOO5Vu38LkH-Cqy955RMn0BKIY0P6jvrI_xXFWz_sWHPs_fOM8Nfjtif34q9j5rutTqOINT-wWu1hRkfsPQ31SHJnN-nZTpGQ27U4vLTAKrn5jsrruw0Bge17VXDYaVoelqCucQUVE/s1600/img24.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjW1eOO5Vu38LkH-Cqy955RMn0BKIY0P6jvrI_xXFWz_sWHPs_fOM8Nfjtif34q9j5rutTqOINT-wWu1hRkfsPQ31SHJnN-nZTpGQ27U4vLTAKrn5jsrruw0Bge17VXDYaVoelqCucQUVE/s320/img24.gif" /></a>y <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjY0nbKw-jBJxqLwCQNRlwT2rZtg9JeoDtPN7DVtP-DsBvJ2sJox37vDUU-xBn-GjuStt3CV8JNyZZtvTXawRHu0ePn7aia7My076pb4i8ljrkv_Tm1NTmjWoAfrGyHpwCL6bpwNBPqfPg/s1600/img25.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjY0nbKw-jBJxqLwCQNRlwT2rZtg9JeoDtPN7DVtP-DsBvJ2sJox37vDUU-xBn-GjuStt3CV8JNyZZtvTXawRHu0ePn7aia7My076pb4i8ljrkv_Tm1NTmjWoAfrGyHpwCL6bpwNBPqfPg/s320/img25.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: center;"> la excentricidad es <br />
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyuk_izwFOAPKXyi0um5roN4RYCbxXiS07RCJ3Pa_BmbDR0yAdnb9lXxuTYmwO-yuA_S8ljwYYul0On8SkLwT-oBMEcNnpAAdWbXkQXALBqsvao-LLPLryDYE4vNwOweWYiwHO0IMDlz8/s1600/img26.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" ox="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyuk_izwFOAPKXyi0um5roN4RYCbxXiS07RCJ3Pa_BmbDR0yAdnb9lXxuTYmwO-yuA_S8ljwYYul0On8SkLwT-oBMEcNnpAAdWbXkQXALBqsvao-LLPLryDYE4vNwOweWYiwHO0IMDlz8/s320/img26.gif" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
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