document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1033118>Question 1033118</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the vertices, the endpoints of the minor axis, and the foci of the given ellipse. \r<BR>\n" );
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document.write( "10x2 + 80x + 7y2 + 42y + 83 = 0<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 #647983 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=647983\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10x%5E2+%2B+80x+%2B+7y%5E2+%2B+42y+%2B+83+=+0+ ALIGN=MIDDLE  ALT=\"10x%5E2+%2B+80x+%2B+7y%5E2+%2B+42y+%2B+83+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%28x%5E2%2B8x%2B16%29%2B7%28y%5E2%2B6y%2B9%29+=+-83%2B160%2B63 ALIGN=MIDDLE  ALT=\"10%28x%5E2%2B8x%2B16%29%2B7%28y%5E2%2B6y%2B9%29+=+-83%2B160%2B63\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%28x%2B4%29%5E2%2B7%28y%2B3%29%5E2+=+140 ALIGN=MIDDLE  ALT=\"10%28x%2B4%29%5E2%2B7%28y%2B3%29%5E2+=+140\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B4%29%5E2%2F14%2B%28y%2B3%29%5E2%2F20+=+1 ALIGN=MIDDLE  ALT=\"%28x%2B4%29%5E2%2F14%2B%28y%2B3%29%5E2%2F20+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "==> the center of the ellipse is the point (-4,-3).\r<BR>\n" );
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document.write( "Now <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2+=+20 ALIGN=MIDDLE  ALT=\"a%5E2+=+20\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+=+14 ALIGN=MIDDLE  ALT=\"b%5E2+=+14\"   DISABLED_style= \"max-width:90%; height:auto;\" >  ==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2+=+20-14+=+6 ALIGN=MIDDLE  ALT=\"c%5E2+=+20-14+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, because for any ellipse, <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( "(Also the major axis is parallel to the y-axis, while the minor axis is parallel to the x-axis.)\r<BR>\n" );
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document.write( "==> the vertices are (-4, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3%2B2sqrt%285%29 ALIGN=MIDDLE  ALT=\"-3%2B2sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and (-4, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3-2sqrt%285%29 ALIGN=MIDDLE  ALT=\"-3-2sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >),<BR>\n" );
document.write( "the co-vertices are (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2Bsqrt%2814%29 ALIGN=MIDDLE  ALT=\"-4%2Bsqrt%2814%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >,-3) and (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4-sqrt%2814%29 ALIGN=MIDDLE  ALT=\"-4-sqrt%2814%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >,-3), <BR>\n" );
document.write( "while the foci are (-4, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3%2Bsqrt%286%29 ALIGN=MIDDLE  ALT=\"-3%2Bsqrt%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and (-4, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3-sqrt%286%29 ALIGN=MIDDLE  ALT=\"-3-sqrt%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >). </SPAN>\n" );
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