document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1074902>Question 1074902</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the foci of the ellipse whose major axis has endpoints $(0,0)$ and $(13,0)$ and whose minor axis has length 12.\r<BR>\n" );
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document.write( "Enter your answer as a list of pairs separated by commas. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #689707 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=689707\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The center of that ellipse is the midpoint of the major axis,<BR>\n" );
document.write( "half way between (0,0) and (13,0) on line y=0.<BR>\n" );
document.write( "The coordinates of that midpoint are the averages of the coordinates of the vertices.<BR>\n" );
document.write( "The midpoint is (6.5,0) , and that is the center of the ellipse.<BR>\n" );
document.write( "The distance from the center to each vertex is called the semi-major axis and represented as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In this case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=6.5 ALIGN=MIDDLE  ALT=\"a=6.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Half of the minor axis is called the semi-minor axis and is represented with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That is the distance from the center to each co-vertex.<BR>\n" );
document.write( "In this case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=12%2F2=6 ALIGN=MIDDLE  ALT=\"b=12%2F2=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The distance from the center to each focus is represented with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and is related to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2=b%5E2%2Bc%5E2 ALIGN=MIDDLE  ALT=\"a%5E2=b%5E2%2Bc%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In this case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6.5%5E2=6%5E2%2Bc%5E2 ALIGN=MIDDLE  ALT=\"6.5%5E2=6%5E2%2Bc%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Solving for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6.5%5E2-6%5E2=c%5E2 ALIGN=MIDDLE  ALT=\"6.5%5E2-6%5E2=c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "A calculator can do that, but I don't need one<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286.5%2B6%29%29%286.5-6%29=c%5E2 ALIGN=MIDDLE  ALT=\"%286.5%2B6%29%29%286.5-6%29=c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.5%2A12.5=c%5E2 ALIGN=MIDDLE  ALT=\"0.5%2A12.5=c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6.25=c%5E2 ALIGN=MIDDLE  ALT=\"6.25=c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=sqrt%286.25%29 ALIGN=MIDDLE  ALT=\"c=sqrt%286.25%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=2.5 ALIGN=MIDDLE  ALT=\"c=2.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That is the distance from center (6.5,0) to each focus.<BR>\n" );
document.write( "The foci are on the same line as the vertices, line y=0,<BR>\n" );
document.write( "on either side of the center,<BR>\n" );
document.write( "so the x-coordinate of one focus is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6.5-2.5=4 ALIGN=MIDDLE  ALT=\"6.5-2.5=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and the x-coordinate of the other focus is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6.5%2B2.5=9 ALIGN=MIDDLE  ALT=\"6.5%2B2.5=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, the foci are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28%22%28+4+%2C+0+%29+%2C+%289+%2C+0+%29%22%29 ALIGN=MIDDLE  ALT=\"highlight%28%22%28+4+%2C+0+%29+%2C+%289+%2C+0+%29%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );