document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=97784>Question 97784</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the vertices and foci of the ellipse given by:\r<BR>\n" );
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document.write( "      x^2+9y^2-24x+18y+9=0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #71159 by </B><A HREF=/tutors/aboutme.mpl?userid=mathslover><B>mathslover(157)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mathslover><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=mathslover><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=71159\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The standard form of the equation of an ellipse is \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28x-h%29%5E2%2Fa%5E2++%2B+%28y-k%29%5E2%2Fb%5E2+=1+ ALIGN=MIDDLE  ALT=\"+%28x-h%29%5E2%2Fa%5E2++%2B+%28y-k%29%5E2%2Fb%5E2+=1+\"   DISABLED_style= \"max-width:90%; height:auto;\" > where (h,k) is the center of the ellipse\r<BR>\n" );
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document.write( "we first try and put the given equation in the standard form \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B9y%5E2-24x%2B18y%2B9=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B9y%5E2-24x%2B18y%2B9=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=+x%5E2+-24x+%2B+144++%2B+9y%5E2+%2B+18y+%2B+9+-144+=0 ALIGN=MIDDLE  ALT=\"+x%5E2+-24x+%2B+144++%2B+9y%5E2+%2B+18y+%2B+9+-144+=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=+x%5E2+-24x+%2B+144++%2B+9%28y%5E2+%2B+2y+%2B+1%29+=144 ALIGN=MIDDLE  ALT=\"+x%5E2+-24x+%2B+144++%2B+9%28y%5E2+%2B+2y+%2B+1%29+=144\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-12%29%5E2+%2B+9%28y+%2B+1%29%5E2+=144+ ALIGN=MIDDLE  ALT=\"%28x-12%29%5E2+%2B+9%28y+%2B+1%29%5E2+=144+\"   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=%28%28x-12%29%5E2%29%2F144+%2B+%289%28y+%2B+1%29%5E2%29%2F144+=1+ ALIGN=MIDDLE  ALT=\"%28%28x-12%29%5E2%29%2F144+%2B+%289%28y+%2B+1%29%5E2%29%2F144+=1+\"   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=%28%28x-12%29%5E2%29%2F144+%2B+%28y+%2B+1%29%5E2%2F16+=1+ ALIGN=MIDDLE  ALT=\"%28%28x-12%29%5E2%29%2F144+%2B+%28y+%2B+1%29%5E2%2F16+=1+\"   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=%28%28x-12%29%5E2%29%2F12%5E2+%2B+%28y+-+%28-1%29%29%5E2%2F4%5E2+=1+ ALIGN=MIDDLE  ALT=\"%28%28x-12%29%5E2%29%2F12%5E2+%2B+%28y+-+%28-1%29%29%5E2%2F4%5E2+=1+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so we have the center (h,k) as (12,-1)\r<BR>\n" );
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document.write( "also a=12 and b=4 <BR>\n" );
document.write( "Since a > b the major axis is horizontal\r<BR>\n" );
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document.write( "vertices are given by (h-a,k) and (h+a,k)<BR>\n" );
document.write( "so we have ( 12-12,-1) and (12+12,-1)<BR>\n" );
document.write( "vertices are (0,-1) and (24,-1)\r<BR>\n" );
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document.write( "focii are at points  given by (h-c,k) and (h+c,k) where c = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28a%5E2+-b%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28a%5E2+-b%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(12- 8*sqrt2,-1)  and (12+ 8*sqrt2,-1)\r<BR>\n" );
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document.write( " </SPAN>\n" );
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