tag:blogger.com,1999:blog-29958134104761166542024-03-13T14:12:39.525-07:00geometrianaliticaSolución de ejercicios y problemas de geometría analítica, por Juan BeltranJuan Beltranhttp://www.blogger.com/profile/03539619598950169589noreply@blogger.comBlogger50125tag:blogger.com,1999:blog-2995813410476116654.post-22683460590011725962014-09-29T13:33:00.000-07:002014-09-29T13:33:04.321-07:00Ecuación de una cuerda de la circunferencia x^2+y^2=25. Lehmann 15.11<div class="separator" style="clear: both; text-align: left;">
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<br />Juan Beltranhttp://www.blogger.com/profile/03539619598950169589noreply@blogger.com1tag:blogger.com,1999:blog-2995813410476116654.post-15590149717518194282014-09-29T10:04:00.003-07:002014-09-29T10:04:26.299-07:00Longitud de una cuerda de la circunferencia x^2+y^2=25. Lehmann 15.10<div class="separator" style="clear: both; text-align: left;">
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<br />Juan Beltranhttp://www.blogger.com/profile/03539619598950169589noreply@blogger.com0tag:blogger.com,1999:blog-2995813410476116654.post-76892337675284581262014-09-29T07:16:00.001-07:002014-09-29T07:16:30.501-07:00Ecuación de la circunferencia dado uno de sus puntos y con centro en la intersección de dos rectas. Lehmann 15.9<div class="separator" style="clear: both; text-align: left;">
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<br />Juan Beltranhttp://www.blogger.com/profile/03539619598950169589noreply@blogger.com0tag:blogger.com,1999:blog-2995813410476116654.post-63661457418733125112014-09-28T08:27:00.004-07:002014-09-28T08:27:48.581-07:00Ecuación de la circunferencia dado el radio y con centro en la intersección de dos rectas. Lehmann 15.8<div class="separator" style="clear: both; text-align: left;">
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<br />Juan Beltranhttp://www.blogger.com/profile/03539619598950169589noreply@blogger.com0tag:blogger.com,1999:blog-2995813410476116654.post-20856359789835216052014-09-27T09:51:00.002-07:002014-09-27T09:51:32.543-07:00Demostrar que de dos puntos dados, uno es interior y el otro exterior a una circunferencia también dada. Lehmann 15_7<div class="separator" style="clear: both; text-align: left;">
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<br />Juan Beltranhttp://www.blogger.com/profile/03539619598950169589noreply@blogger.com1tag:blogger.com,1999:blog-2995813410476116654.post-60434692995378312922014-09-27T07:21:00.003-07:002014-09-27T07:21:49.478-07:00Ecuación de la circunferencia dadas las coordenadas del centro y tangente a una recta dada. Lehmann 15.5<div class="separator" style="clear: both; text-align: left;">
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<br />Juan Beltranhttp://www.blogger.com/profile/03539619598950169589noreply@blogger.com0tag:blogger.com,1999:blog-2995813410476116654.post-40426275925771212112014-09-26T15:50:00.001-07:002014-09-26T15:50:01.638-07:00Ecuación de la circunferencia dadas las coordenadas del centro y uno de sus puntos. Lehmann 15.3<div class="separator" style="clear: both; text-align: left;">
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<br />Juan Beltranhttp://www.blogger.com/profile/03539619598950169589noreply@blogger.com0tag:blogger.com,1999:blog-2995813410476116654.post-39657748285328412282014-09-26T15:29:00.004-07:002014-09-26T15:29:55.672-07:00Ecuación de la circunferencia dados los extremos de un diámetro. Lehmann 15.2<div class="separator" style="clear: both; text-align: left;">
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<br />Juan Beltranhttp://www.blogger.com/profile/03539619598950169589noreply@blogger.com4tag:blogger.com,1999:blog-2995813410476116654.post-24279124527078904572014-09-26T15:09:00.003-07:002014-09-26T15:09:57.906-07:00Ecuación de la circunferencia. Centro y radio. Lehmann 15.1<div class="separator" style="clear: both; text-align: left;">
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<br />Juan Beltranhttp://www.blogger.com/profile/03539619598950169589noreply@blogger.com0tag:blogger.com,1999:blog-2995813410476116654.post-91018011282600601072014-09-24T10:54:00.004-07:002014-09-24T10:55:42.178-07:00Ecuación de la recta que pasa por el punto de intersección de dos rectas dadas y que a su vez es perpendicular a una tercera recta también dada.<div class="separator" style="clear: both; text-align: left;">
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<br />Juan Beltranhttp://www.blogger.com/profile/03539619598950169589noreply@blogger.com11tag:blogger.com,1999:blog-2995813410476116654.post-89449823975448227572014-09-24T06:49:00.005-07:002014-09-24T08:24:30.716-07:00Circunferencia circunscrita a un triángulo. Baricentro.<div class="separator" style="clear: both; text-align: left;">
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<br />Juan Beltranhttp://www.blogger.com/profile/03539619598950169589noreply@blogger.com0tag:blogger.com,1999:blog-2995813410476116654.post-67518234987954613562013-07-07T15:49:00.000-07:002014-10-02T13:35:41.262-07:00Ecuación simétrica de la recta. Lehmann 9.7<span style="font-family: Georgia,"Times New Roman",serif;"><span style="font-size: medium;"><span style="color: #cc0000;">9.7</span> Una recta pasa por los dos puntos A(-3,-1) y B(2,-6). Hallar la ecuación en la forma simétrica.</span></span><br />
<span style="font-family: Georgia,"Times New Roman",serif;"><span style="color: #6aa84f;"><span style="font-size: medium;">Solución-Juan Beltrán:</span></span></span><br />
<span style="font-family: Georgia,"Times New Roman",serif;"><span style="color: #6aa84f;"><span style="font-size: medium;"><span style="color: black;"><span style="color: blue;">Teorema:</span> La recta que pasa por dos puntos dados P1(x1, y1) y P2(x2,y2) tiene por ecuación:</span></span></span></span><br />
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<span style="font-family: Georgia,"Times New Roman",serif; font-size: medium;"><span style="color: #6aa84f;"><span style="font-size: medium;"><span style="color: black;"><span style="color: blue;"><br /></span></span></span></span></span><br />
<span style="font-family: Georgia,"Times New Roman",serif; font-size: medium;"><span style="color: #6aa84f;"><span style="font-size: medium;"><span style="color: black;"><span style="color: blue;">Teorema:</span> La recta cuyas intercepciones con los ejes X y Y son <i>a</i> y <i>b</i> respectivamente, <i>a</i> y <i>b</i> diferentes de 0, </span></span></span></span><span style="font-family: Georgia, 'Times New Roman', serif; font-size: medium;">tiene por ecuación:</span><br />
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<br />Juan Beltranhttp://www.blogger.com/profile/03539619598950169589noreply@blogger.com10tag:blogger.com,1999:blog-2995813410476116654.post-74296909995784018202013-06-26T10:18:00.002-07:002013-07-07T14:52:21.836-07:00Pendiente y ángulo de inclinación de una recta que pasa por dos puntos dados. Lehmann 3.4<!--[if gte mso 9]><xml>
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<![endif]--><span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;"><span style="color: #cc0000;">3.4</span> Hallar la pendiente y el ángulo de
inclinación de la recta que pasa por los puntos (-3, 2) y (7, -3).</span></span><br />
<span style="font-family: Times,"Times New Roman",serif;"><span style="color: #6aa84f;"><span style="font-size: large;">Solución-Juan Beltrán:</span></span></span><br />
<div class="MsoNormal">
<span style="font-size: large;"><span lang="ES-MX" style="font-family: "Times New Roman","serif";"><span style="color: blue;">Definición:</span> el <span style="color: purple;"><i>ángulo de inclinación</i></span> de una recta en
el plano es aquél formado por el semieje positivo <i>X</i> y la recta.</span></span></div>
<span style="font-size: large;">
</span>
<br />
<div class="MsoNormal">
<span style="font-size: large;"><span lang="ES-MX" style="font-family: "Times New Roman","serif";"><span style="color: blue;">Definición:</span> la <span style="color: purple;"><i>pendiente</i></span> o coeficiente angular, <i>m</i>, de una recta es la tangente del ángulo de
inclinación de la recta. Si <i>A</i> es el ángulo de inclinación, se tiene entonces
que:</span></span></div>
<div class="MsoNormal" style="text-align: center;">
<span style="color: red;"><span style="font-size: large;"><span lang="ES-MX" style="font-family: "Times New Roman","serif";"><i>m =</i> tan<i>A</i></span></span></span></div>
<div class="MsoNormal">
<span style="font-size: large;"><span lang="ES-MX" style="font-family: "Times New Roman","serif";"><span style="color: blue;">Teorema:</span> Si <i>P</i><sub>1</sub>(<i>x</i><sub>1</sub>, <i>y</i><sub>1</sub>)
y <i>P</i><sub>2</sub>(<i>x</i><sub>2</sub>, <i>y</i><sub>2</sub>) son dos puntos distintos de una recta, la
pendiente, <i>m</i>, de la recta está dada por: </span></span><br />
<div class="separator" style="clear: both; text-align: center;">
<span style="font-size: large;"><span lang="ES-MX" style="font-family: "Times New Roman","serif";"><a href="http://1.bp.blogspot.com/-SgXpa1o_Mus/UcseM5u0j2I/AAAAAAAAALo/R1Har-ZOZzo/s1600/Pendiente.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-SgXpa1o_Mus/UcseM5u0j2I/AAAAAAAAALo/R1Har-ZOZzo/s1600/Pendiente.png" /></a></span></span></div>
<div class="separator" style="clear: both; text-align: center;">
<span style="font-size: large;"><span lang="ES-MX" style="font-family: "Times New Roman","serif";"><a href="http://3.bp.blogspot.com/-nCQ9ULKPe2s/UcsiMPEqkYI/AAAAAAAAAMA/dYWNTDpVGLU/s1600/3.4.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-nCQ9ULKPe2s/UcsiMPEqkYI/AAAAAAAAAMA/dYWNTDpVGLU/s1600/3.4.png" /></a></span></span></div>
</div>
Juan Beltranhttp://www.blogger.com/profile/03539619598950169589noreply@blogger.com27tag:blogger.com,1999:blog-2995813410476116654.post-14364046621433701212013-06-25T08:44:00.000-07:002013-06-25T10:59:43.630-07:00Translación de ejes. Transformar una ecuación. Lehmann 20.7.<span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;"><span style="color: #6aa84f;">Teorema:</span> Si se trasladan los ejes coordenados a un nuevo origen O'(<i>h</i>, <i>k</i>), y si las coordenadas de cualquier punto <i>P</i> antes y después de la traslación son (<i>x</i>, <i>y</i>) y (<i>x</i>', <i>y</i>'), respectivamente, las ecuaciones de transformación del sistema primitivo al nuevo sistema de coordenadas son: </span></span><br />
<div style="text-align: center;">
<span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;"><i><span style="color: #cc0000;">x = x' + h </span></i></span></span></div>
<div style="text-align: center;">
<span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;"><i><span style="color: #cc0000;">y = y' + k </span></i></span></span></div>
<div style="text-align: left;">
</div>
<div class="MsoNormal" style="text-align: left;">
<span style="font-size: large;"><span lang="ES-MX" style="font-family: "Times New Roman","serif";"><span style="color: #cc0000;">20.7</span> Por una traslación de ejes, transforme la
ecuación 3<i>x</i><sup>2</sup> + 2<i>y</i><sup>2 </sup>+ 18<i>x </i>- 8<i>y</i> + 29 = 0 en otra
ecuación que carezca de términos de primer grado.</span></span></div>
<div class="MsoNormal" style="text-align: left;">
<span style="font-size: large;"><span lang="ES-MX" style="font-family: "Times New Roman","serif";"><span style="color: #6aa84f;">Solución-Juan Beltran</span>:</span></span><br />
<div class="separator" style="clear: both; text-align: center;">
<a href="http://4.bp.blogspot.com/-v00TKWJ01QE/Ucm6V7wfSRI/AAAAAAAAALU/GHE6_SlPCu4/s1600/20_7.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="396" src="http://4.bp.blogspot.com/-v00TKWJ01QE/Ucm6V7wfSRI/AAAAAAAAALU/GHE6_SlPCu4/s528/20_7.png" width="528" /></a></div>
</div>
Juan Beltranhttp://www.blogger.com/profile/03539619598950169589noreply@blogger.com4tag:blogger.com,1999:blog-2995813410476116654.post-19438361989648969332013-06-25T06:41:00.001-07:002013-06-25T06:41:18.316-07:00Ecuación de la hipérbola. Coordenadas de los vértices y focos, las longitudes de los ejes transverso y conjugado, la excentricidad y la longitud de cada lado recto. Lehmann 30.6<div class="MsoNormal">
<span style="font-size: large;"><span style="font-family: Times,"Times New Roman",serif;"><span lang="ES-MX"><span style="color: #cc0000;">30.6</span></span><span style="color: #cc0000;"> </span>Hállense las coordenadas de los vértices y
focos, las longitudes de los ejes transverso y conjugado, la excentricidad y la
longitud de cada lado recto. Trácese y discútase el lugar geométrico. Para la
ecuación 9<i>x</i><span style="font-size: small;"><sup>2</sup></span>-4<i>y</i><span style="font-size: small;"><sup>2</sup></span>=36.</span></span></div>
<div class="MsoNormal">
<span style="font-size: large;"><span style="font-family: Times,"Times New Roman",serif;"><span lang="ES-MX"><span style="color: #38761d;">Solución-Juan Beltran:</span> </span></span></span></div>
<span style="font-size: large;"><span style="font-family: Times,"Times New Roman",serif;">
<span style="color: blue;">Definición:</span> una <span style="color: #e69138;"><i>hipérbola</i></span>
es el lugar geométrico de un punto que se mueve en el plano de tal
manera que el valor absoluto de la diferencia de sus distancias a dos
puntos fijos del plano, llamados focos, es siempre constante, positiva y
menor que la distancia entre los focos.</span></span><br />
<div class="separator" style="clear: both; text-align: center;">
<a href="http://4.bp.blogspot.com/-VoNx6ynzQ-c/UcmdzxasM4I/AAAAAAAAALA/B_mRc_uk02U/s1600/30.6_.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="396" src="http://4.bp.blogspot.com/-VoNx6ynzQ-c/UcmdzxasM4I/AAAAAAAAALA/B_mRc_uk02U/s528/30.6_.png" width="528" /></a></div>
Juan Beltranhttp://www.blogger.com/profile/03539619598950169589noreply@blogger.com0tag:blogger.com,1999:blog-2995813410476116654.post-63072260239048240862013-06-25T06:30:00.003-07:002013-06-25T06:31:16.405-07:00Ecuación de la hipérbola. Lehmann 30.2<div class="MsoNormal">
<span style="font-size: large;"><span lang="ES-MX" style="font-family: "Times New Roman","serif";"><span style="color: #cc0000;">30.2</span> Demostrar que si <i>P</i><sub>1 </sub>es un punto cualquiera cuyas coordenadas satisfacen
la ecuación <i>b</i><sup>2</sup><i>x</i><sup>2</sup></span><span lang="ES-MX" style="font-family: Symbol;">-</span><i><span lang="ES-MX" style="font-family: "Times New Roman","serif";">a</span></i><sup><span lang="ES-MX" style="font-family: "Times New Roman","serif";">2</span></sup><i><span lang="ES-MX" style="font-family: "Times New Roman","serif";">y</span></i><sup><span lang="ES-MX" style="font-family: "Times New Roman","serif";">2</span></sup><span lang="ES-MX" style="font-family: Symbol;">=</span><i><span lang="ES-MX" style="font-family: "Times New Roman","serif";">a</span></i><sup><span lang="ES-MX" style="font-family: "Times New Roman","serif";">2</span></sup><i><span lang="ES-MX" style="font-family: "Times New Roman","serif";">b</span></i><sup><span lang="ES-MX" style="font-family: "Times New Roman","serif";">2</span></sup><span lang="ES-MX" style="font-family: "Times New Roman","serif";">,
entonces <i>P</i><sub>1</sub> está sobre la
hipérbola representada por esta ecuación.</span></span></div>
<div class="MsoNormal">
<span style="font-size: large;"><span lang="ES-MX" style="font-family: "Times New Roman","serif";"><span style="color: #38761d;">Solución-Juan Beltran:</span> </span></span></div>
<span style="font-size: large;">
</span><span style="font-size: large;"><span style="color: blue;">Definición:</span> una <span style="color: #e69138;"><i>hipérbola</i></span>
es el lugar geométrico de un punto que se mueve en el plano de tal
manera que el valor absoluto de la diferencia de sus distancias a dos
puntos fijos del plano, llamados focos, es siempre constante, positiva y
menor que la distancia entre los focos.</span><br />
<div class="separator" style="clear: both; text-align: center;">
<a href="http://2.bp.blogspot.com/-R4z1XU9ppU0/UcmbTZe87TI/AAAAAAAAAKw/_xOTZq4Ltdo/s1600/30.2_2.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="200" src="http://2.bp.blogspot.com/-R4z1XU9ppU0/UcmbTZe87TI/AAAAAAAAAKw/_xOTZq4Ltdo/s528/30.2_2.png" width="528" /></a></div>
Juan Beltranhttp://www.blogger.com/profile/03539619598950169589noreply@blogger.com0tag:blogger.com,1999:blog-2995813410476116654.post-59308515627128537752013-06-25T05:42:00.000-07:002013-06-25T05:42:20.774-07:00Ecuación de la elipse. Lehmann 27.29<span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;"><span style="color: #cc0000;">27.29</span>
Hallar e identificar la ecuación del lugar geométrico de los puntos que dividen a las ordenadas de los puntos de la circunferencia <i>x</i>^2+<i>y</i>^2=16 en la razón 1:4.</span></span><br />
<span style="color: #38761d;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;">Solución-Juan Beltrán:</span></span></span><br />
<span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;"><span style="color: blue;">Definición:</span> Una elipse es el lugar geométrico de un punto <i>P</i> que se mueve en el plano de tal modo que la suma de sus distancias a dos puntos fijos, llamados focos de la elipse, <i>F</i> y <i>F'</i>, de ese plano es constante y mayor que la distancia entre los dos focos.</span></span><br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;"><a href="http://1.bp.blogspot.com/-qg_9oqnd-xI/UcmP1qrUGVI/AAAAAAAAAKY/V6kFjigcHWs/s1600/27.29.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="450" src="http://1.bp.blogspot.com/-qg_9oqnd-xI/UcmP1qrUGVI/AAAAAAAAAKY/V6kFjigcHWs/s528/27.29.png" width="528" /></a></span></span></div>
<a href="http://www.geometrianalitica.calculo21.org/id490.htm">Más soluciones de los ejercicios de Lehmann AQUI
</a>
Juan Beltranhttp://www.blogger.com/profile/03539619598950169589noreply@blogger.com2tag:blogger.com,1999:blog-2995813410476116654.post-28582825637206898852013-06-25T05:33:00.000-07:002013-06-25T05:33:34.690-07:00Ecuación de la elipse. Lehmann 27.27<span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;"><span style="color: #cc0000;">27.27</span>
Hallar e identificar la ecuación del lugar geométrico de un punto que se nueve de tal manera que su distancia a la recta <i>y</i>=-8 es siempre igual al doble de su distancia del punto (0,-2)</span></span><br />
<span style="color: #38761d;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;">Solución-Juan Beltrán:</span></span></span><br />
<span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;"><span style="color: blue;">Definición:</span> Una elipse es el lugar geométrico de un punto <i>P</i> que se mueve en el plano de tal modo que la suma de sus distancias a dos puntos fijos, llamados focos de la elipse, <i>F</i> y <i>F'</i>, de ese plano es constante y mayor que la distancia entre los dos focos.</span></span><br />
<div class="separator" style="clear: both; text-align: center;">
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<br />
<a href="http://www.geometrianalitica.calculo21.org/id490.htm">Más soluciones de los ejercicios de Lehmann AQUI
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Juan Beltranhttp://www.blogger.com/profile/03539619598950169589noreply@blogger.com0tag:blogger.com,1999:blog-2995813410476116654.post-44602650285699565072013-06-24T06:19:00.001-07:002013-06-24T06:24:01.044-07:00Ecuación de la elipse, coordenadas de los vértices y focos, las longitudes de los ejes mayor y menor, la excentricidad y la longitud de los lados rectos Lehmann 27.6<span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;"><span style="color: #cc0000;">27.6</span> Hallar las coordenadas de los vértices y focos, las longitudes de los ejes mayor y menor, la excentricidad y la longitud de cada uno de los lados rectos de la elipse cuya ecuación es 9<i>x^</i>2<i>+</i>4<i>y^</i>2<i>=</i>36.</span></span><br />
<span style="color: #38761d;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;">Solución-Juan Beltrán:</span></span></span><br />
<span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;"><span style="color: blue;">Definición:</span> Una elipse es el lugar geométrico de un punto <i>P</i> que se mueve en el plano de tal modo que la suma de sus distancias a dos puntos fijos, llamados focos de la elipse, <i>F</i> y <i>F'</i>, de ese plano es constante y mayor que la distancia entre los dos focos.</span></span><br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;"><a href="http://3.bp.blogspot.com/-sLfnOhIOADI/UchGiTPI_1I/AAAAAAAAAJ4/iRjOoh6irZc/s1600/27.6.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="396" src="http://3.bp.blogspot.com/-sLfnOhIOADI/UchGiTPI_1I/AAAAAAAAAJ4/iRjOoh6irZc/s528/27.6.png" width="528" /></a></span></span></div>
<a href="http://www.geometrianalitica.calculo21.org/id490.htm">Más soluciones de los ejercicios de Lehmann AQUI
</a>Juan Beltranhttp://www.blogger.com/profile/03539619598950169589noreply@blogger.com3tag:blogger.com,1999:blog-2995813410476116654.post-56610277846385175712013-06-24T06:03:00.000-07:002013-06-24T06:03:51.770-07:00Ecuación de la elipse (forma ordinaria). Lehmann 27.1<span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;"><span style="color: #cc0000;">27.1</span> Deducir la ecuación ordinaria (canónica) de la elipse en el caso que el eje focal coincida con el eje<i>y</i>.</span></span><br />
<span style="color: #38761d;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;">Solución-Juan Beltrán:</span></span></span><br />
<span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;"><span style="color: blue;">Definición:</span> Una elipse es el lugar geométrico de un punto <i>P</i> que se mueve en el plano de tal modo que la suma de sus distancias a dos puntos fijos, llamados focos de la elipse, <i>F</i> y <i>F'</i>, de ese plano es constante y mayor que la distancia entre los dos focos.</span></span><br />
<div class="separator" style="clear: both; text-align: center;">
<a href="http://3.bp.blogspot.com/-IbrFdOaRF3Q/UchDUekuUPI/AAAAAAAAAJo/Nri4j5K2LLc/s1600/27.1.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="528" src="http://3.bp.blogspot.com/-IbrFdOaRF3Q/UchDUekuUPI/AAAAAAAAAJo/Nri4j5K2LLc/s528/27.1.png" width="520" /></a></div>
<br />Juan Beltranhttp://www.blogger.com/profile/03539619598950169589noreply@blogger.com0tag:blogger.com,1999:blog-2995813410476116654.post-9327808078833651742013-06-22T06:08:00.002-07:002013-06-22T06:08:23.903-07:00Ecuación de la parábola de vértice en el origen y eje un eje coordenado. Lehmann 23.2<span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;">“La <span style="color: #990000;">parábola</span>
se define como el lugar geométrico de los puntos de un plano que
equidistan de una recta llamada directriz, y un punto exterior a ella
llamado foco”.
<a href="http://es.wikipedia.org/wiki/Par%C3%A1bola_%28matem%C3%A1tica%29">Más información</a>.</span></span><br />
<br />
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Juan Beltranhttp://www.blogger.com/profile/03539619598950169589noreply@blogger.com0tag:blogger.com,1999:blog-2995813410476116654.post-20131184717855812532013-06-22T06:06:00.001-07:002013-06-22T06:06:26.959-07:00Ecuación de la parábola de vértice en el origen y eje un eje coordenado. Lehmann 23.1<span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;">“La <span style="color: #990000;">parábola</span> se define como el lugar geométrico de los puntos de un plano que equidistan de una recta llamada directriz, y un punto exterior a ella llamado foco”.
<a href="http://es.wikipedia.org/wiki/Par%C3%A1bola_%28matem%C3%A1tica%29">Más información</a>.</span></span><br />
<br />
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Juan Beltranhttp://www.blogger.com/profile/03539619598950169589noreply@blogger.com0tag:blogger.com,1999:blog-2995813410476116654.post-39470962095270662962013-06-22T05:46:00.000-07:002013-06-22T05:46:07.306-07:00Transformación de coordenadas. Traslación de los ejes coordenados. Lehmann 20.5<span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;"><span style="color: #6aa84f;"><b>Teorema:</b></span> Si se trasladan los ejes coordenados a un nuevo origen O'(<i>h</i>, <i>k</i>), y si las coordenadas de cualquier punto <i>P</i> antes y después de la traslación son (<i>x</i>, y) y (<i>x</i>', <i>y</i>'), respectivamente, las ecuaciones de transformación del sistema primitivo al nuevo sistema de coordenadas son: </span></span><br />
<div style="text-align: center;">
<span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;"><i><span style="color: #cc0000;">x = x' + h </span></i></span></span></div>
<div style="text-align: center;">
<span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;"><i><span style="color: #cc0000;">y = y' + k </span></i></span></span></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="http://3.bp.blogspot.com/-gq2M9n9fvk8/UcWbqcv9CCI/AAAAAAAAAI4/5Mjk_gdctEs/s1600/20_5.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="528" src="http://3.bp.blogspot.com/-gq2M9n9fvk8/UcWbqcv9CCI/AAAAAAAAAI4/5Mjk_gdctEs/s528/20_5.png" width="525" /></a></div>
<br />Juan Beltranhttp://www.blogger.com/profile/03539619598950169589noreply@blogger.com0tag:blogger.com,1999:blog-2995813410476116654.post-82126467802079713082013-06-22T05:40:00.000-07:002013-06-22T05:41:34.201-07:00Transformación de coordenadas. Traslación de los ejes coordenados. Lehmann 20.3<span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;"><span style="color: #45818e;"><b>Teorema:</b></span> Si se trasladan los ejes coordenados a un nuevo origen O'(<i>h</i>, <i>k</i>), y si las coordenadas de cualquier punto <i>P</i> antes y después de la traslación son (<i>x</i>, <i>y</i>) y (<i>x</i>', <i>y</i>'), respectivamente, las ecuaciones de transformación del sistema primitivo al nuevo sistema de coordenadas son: </span></span><br />
<div style="text-align: center;">
<span style="color: #cc0000;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;"><i>x</i> = <i>x</i>' + <i>h</i> </span></span></span></div>
<div style="text-align: center;">
<span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;"><span style="color: #cc0000;"><i>y</i> = <i>y</i>' + <i>k</i></span> </span></span></div>
<div style="text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://2.bp.blogspot.com/-fPy8dDQyREM/UcWaIoHEdmI/AAAAAAAAAIo/LZTTEe9jQtI/s1600/20_3.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="528" src="http://2.bp.blogspot.com/-fPy8dDQyREM/UcWaIoHEdmI/AAAAAAAAAIo/LZTTEe9jQtI/s528/20_3.png" width="510" /></a></div>
<br />Juan Beltranhttp://www.blogger.com/profile/03539619598950169589noreply@blogger.com1tag:blogger.com,1999:blog-2995813410476116654.post-46982859236119370262013-06-22T05:14:00.000-07:002013-06-22T05:35:06.913-07:00Transformación de coordenadas. Traslación de los ejes coordenados. Lehmann 20.1<span style="font-size: large;"><span style="font-family: Times,"Times New Roman",serif;"><span style="color: #a64d79;"><b>Teorema:</b></span> Si se trasladan los ejes coordenados a un nuevo origen O'(<i>h</i>, <i>k</i>), y si las coordenadas de cualquier punto <i>P </i>antes y después de la traslación son (<i>x</i>,<i> y</i>) y (<i>x</i>', <i>y</i>'), respectivamente, las ecuaciones de transformación del sistema primitivo al nuevo sistema de coordenadas son:</span></span><br />
<div style="text-align: center;">
<span style="color: #990000;"><i><span style="font-size: large;"><span style="font-family: Times,"Times New Roman",serif;">x = x' + h </span></span></i></span></div>
<div style="text-align: center;">
<span style="font-size: large;"><span style="font-family: Times,"Times New Roman",serif;"><span style="color: #990000;"><i>y = y' + k</i></span> </span></span></div>
<div style="text-align: center;">
<span style="font-size: large;"><span style="font-family: Times,"Times New Roman",serif;"> </span></span>
</div>
<div class="separator" style="clear: both; text-align: center;">
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<br />Juan Beltranhttp://www.blogger.com/profile/03539619598950169589noreply@blogger.com0