document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199602>Question 1199602</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the equation of the ellipse with center at (0,0), vertices at (2,0)<BR>\n" );
document.write( "and (-2,0), and eccentricity of 2/7. Sketch the ellipse.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848270 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848270\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Determine the value of 'a'**\r<BR>\n" );
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document.write( "* The vertices are at (±2, 0), which indicates that the major axis lies along the x-axis.<BR>\n" );
document.write( "* The distance from the center (0, 0) to a vertex is the length of the semi-major axis (a).<BR>\n" );
document.write( "* Therefore, a = 2\r<BR>\n" );
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document.write( "**2. Determine the value of 'c'**\r<BR>\n" );
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document.write( "* Eccentricity (e) = c/a <BR>\n" );
document.write( "* where:<BR>\n" );
document.write( "    * e = eccentricity (2/7)<BR>\n" );
document.write( "    * c = distance from the center to a focus <BR>\n" );
document.write( "    * a = semi-major axis (2)\r<BR>\n" );
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document.write( "* 2/7 = c / 2<BR>\n" );
document.write( "* c = 4/7\r<BR>\n" );
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document.write( "**3. Determine the value of 'b'**\r<BR>\n" );
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document.write( "* The relationship between a, b, and c in an ellipse is:<BR>\n" );
document.write( "    * c² = a² - b²\r<BR>\n" );
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document.write( "* (4/7)² = 2² - b²<BR>\n" );
document.write( "* 16/49 = 4 - b²<BR>\n" );
document.write( "* b² = 4 - 16/49 <BR>\n" );
document.write( "* b² = 176/49<BR>\n" );
document.write( "* b = √(176/49) = (4√11) / 7 \r<BR>\n" );
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document.write( "**4. Write the equation of the ellipse**\r<BR>\n" );
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document.write( "* The standard equation of an ellipse centered at the origin with the major axis along the x-axis is:<BR>\n" );
document.write( "    * x²/a² + y²/b² = 1\r<BR>\n" );
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document.write( "* Substitute the values of a² and b²:<BR>\n" );
document.write( "    * x²/2² + y²/[(4√11)/7]² = 1<BR>\n" );
document.write( "    * x²/4 + y²/(176/49) = 1<BR>\n" );
document.write( "    * x²/4 + 49y²/176 = 1\r<BR>\n" );
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document.write( "**Therefore, the equation of the ellipse is x²/4 + 49y²/176 = 1**\r<BR>\n" );
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document.write( "**Sketching the Ellipse**\r<BR>\n" );
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document.write( "1. **Plot the center:** (0, 0)<BR>\n" );
document.write( "2. **Plot the vertices:** (2, 0) and (-2, 0)<BR>\n" );
document.write( "3. **Determine the endpoints of the minor axis:** <BR>\n" );
document.write( "    * Since b = (4√11)/7, the endpoints of the minor axis are (0, (4√11)/7) and (0, -(4√11)/7)<BR>\n" );
document.write( "4. **Sketch the ellipse:** <BR>\n" );
document.write( "    * Draw a smooth curve connecting the vertices and the endpoints of the minor axis.\r<BR>\n" );
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document.write( "**Key Points:**\r<BR>\n" );
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document.write( "* The major axis of the ellipse lies along the x-axis.<BR>\n" );
document.write( "* The eccentricity (2/7) indicates that the ellipse is relatively elongated.\r<BR>\n" );
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document.write( "I hope this helps! Let me know if you have any other questions.<BR>\n" );
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