document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165474>Question 1165474</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Find the equation of the ellipse whose center is at (-3, -1), vertex at (2, -1), and focus at (1, -1).<BR>\n" );
document.write( "a. (9x)2 + (36y)2  -  54x + 50y  -  116 = 0 <BR>\n" );
document.write( "b. (4x)2 + (25y)2 + 54x  -  50y  -  122 = 0 <BR>\n" );
document.write( "c. (9x)2 + (25y)2 + 50x + 50y + 109 = 0<BR>\n" );
document.write( "d. (9x)2 + (25y)2 + 54x + 50y  -  119 = 0  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #790021 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=790021\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the equation of the 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;\" >\r<BR>\n" );
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document.write( "given:\r<BR>\n" );
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document.write( "center is at (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >)=> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-3 ALIGN=MIDDLE  ALT=\"h=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-1 ALIGN=MIDDLE  ALT=\"k=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so far we have:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%5E2%2Fa%5E2+%2B+%28y%2B1%29%5E2%2Fb%5E2+=+1 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%5E2%2Fa%5E2+%2B+%28y%2B1%29%5E2%2Fb%5E2+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "focus at (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-1 ALIGN=MIDDLE  ALT=\"+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >) <BR>\n" );
document.write( "vertex at (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >) \r<BR>\n" );
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document.write( "The major axis a is the segment that contains both foci  and has its endpoints on the ellipse. These endpoints are called the vertices. The <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=midpoint ALIGN=MIDDLE  ALT=\"midpoint\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the major axis is the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=center ALIGN=MIDDLE  ALT=\"center\"   DISABLED_style= \"max-width:90%; height:auto;\" >  of the ellipse.\r<BR>\n" );
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document.write( "so using midpoint formula,  the other vertex will be  at \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2Bx%29%2F2=-3 ALIGN=MIDDLE  ALT=\"%282%2Bx%29%2F2=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2%2Bx=-6 ALIGN=MIDDLE  ALT=\"+2%2Bx=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x=-6-2 ALIGN=MIDDLE  ALT=\"+x=-6-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >=> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-8 ALIGN=MIDDLE  ALT=\"x=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-1%2By%29%2F2=-1 ALIGN=MIDDLE  ALT=\"%28-1%2By%29%2F2=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2By=-2 ALIGN=MIDDLE  ALT=\"-1%2By=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2%2B1 ALIGN=MIDDLE  ALT=\"y=-2%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >=><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-1 ALIGN=MIDDLE  ALT=\"y=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "other vertex will be  at (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8 ALIGN=MIDDLE  ALT=\"-8\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "and the major axis =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a ALIGN=MIDDLE  ALT=\"2a\"   DISABLED_style= \"max-width:90%; height:auto;\" > is distance between  vertices\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a=sqrt%28%282-%28-8%29%29%5E2%2B%28-1-%28-1%29%29%5E2%29 ALIGN=MIDDLE  ALT=\"2a=sqrt%28%282-%28-8%29%29%5E2%2B%28-1-%28-1%29%29%5E2%29\"   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=2a=sqrt%2810%5E2%2B0%5E2%29 ALIGN=MIDDLE  ALT=\"2a=sqrt%2810%5E2%2B0%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a=10 ALIGN=MIDDLE  ALT=\"2a=10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=5 ALIGN=MIDDLE  ALT=\"a=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so far \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%5E2%2F5%5E2+%2B+%28y%2B1%29%5E2%2Fb%5E2+=+1 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%5E2%2F5%5E2+%2B+%28y%2B1%29%5E2%2Fb%5E2+=+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=%28x%2B3%29%5E2%2F25+%2B+%28y%2B1%29%5E2%2Fb%5E2+=+1 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%5E2%2F25+%2B+%28y%2B1%29%5E2%2Fb%5E2+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The formula generally associated with the focus of an ellipse is<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+c%5E2=a%5E2-b%5E2 ALIGN=MIDDLE  ALT=\"+c%5E2=a%5E2-b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+ ALIGN=MIDDLE  ALT=\"c+\"   DISABLED_style= \"max-width:90%; height:auto;\" >is the distance from the focus to center , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+ ALIGN=MIDDLE  ALT=\"a+\"   DISABLED_style= \"max-width:90%; height:auto;\" >is the distance from the center to a vetex 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;\" >is the distance from the center to a co-vetex\r<BR>\n" );
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document.write( "if center is at (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and focus at (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >) \r<BR>\n" );
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document.write( " and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the distance from the focus to center, then=>\r<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=formula_distance.solver>Distance Between 2 points</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>The distance formula is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28%28x%5B2%5D-x%5B1%5D%29%5E2%29%2B%28%28y%5B2%5D-y%5B1%5D%29%5E2%29%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28%28x%5B2%5D-x%5B1%5D%29%5E2%29%2B%28%28y%5B2%5D-y%5B1%5D%29%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Plug in the numbers,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28%28-3-%281%29%29%5E2%29%2B%28%28-1-%28-1%29%29%5E2%29%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28%28-3-%281%29%29%5E2%29%2B%28%28-1-%28-1%29%29%5E2%29%29\"   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=sqrt%28-4%5E2%2B0%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28-4%5E2%2B0%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  The distance is 4.\r<BR>\n" );
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document.write( "so, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=5 ALIGN=MIDDLE  ALT=\"a=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=4 ALIGN=MIDDLE  ALT=\"c=4\"   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=c%5E2=a%5E2-b%5E2 ALIGN=MIDDLE  ALT=\"c%5E2=a%5E2-b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2=a%5E2-c%5E2 ALIGN=MIDDLE  ALT=\"b%5E2=a%5E2-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=b%5E2=5%5E2-4%5E2 ALIGN=MIDDLE  ALT=\"b%5E2=5%5E2-4%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2=25-16 ALIGN=MIDDLE  ALT=\"b%5E2=25-16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2=9 ALIGN=MIDDLE  ALT=\"b%5E2=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so, your equation is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%5E2%2F25+%2B+%28y%2B1%29%5E2%2F9+=+1 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%5E2%2F25+%2B+%28y%2B1%29%5E2%2F9+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "expand:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%5E2%2F25+%2B+%28y%2B1%29%5E2%2F9+=+1 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%5E2%2F25+%2B+%28y%2B1%29%5E2%2F9+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >... both sides multiply by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25%2A9=225 ALIGN=MIDDLE  ALT=\"25%2A9=225\"   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=225%28x%2B3%29%5E2%2F25+%2B+225%28y%2B1%29%5E2%2F9+=+225 ALIGN=MIDDLE  ALT=\"225%28x%2B3%29%5E2%2F25+%2B+225%28y%2B1%29%5E2%2F9+=+225\"   DISABLED_style= \"max-width:90%; height:auto;\" >..........simplify\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%28x%2B3%29%5E2+%2B+25%28y%2B1%29%5E2+=+225 ALIGN=MIDDLE  ALT=\"9%28x%2B3%29%5E2+%2B+25%28y%2B1%29%5E2+=+225\"   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=9%28x%5E2+%2B+6+x+%2B+9%29+%2B+25%28y%5E2%2B2y%2B1%29+=+225 ALIGN=MIDDLE  ALT=\"9%28x%5E2+%2B+6+x+%2B+9%29+%2B+25%28y%5E2%2B2y%2B1%29+=+225\"   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=9x%5E2+%2B+54+x+%2B+81+%2B+25y%5E2%2B50y%2B25+=+225 ALIGN=MIDDLE  ALT=\"9x%5E2+%2B+54+x+%2B+81+%2B+25y%5E2%2B50y%2B25+=+225\"   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=9x%5E2+%2B+54+x+%2B+81+%2B+25y%5E2%2B50y%2B25+=+225 ALIGN=MIDDLE  ALT=\"9x%5E2+%2B+54+x+%2B+81+%2B+25y%5E2%2B50y%2B25+=+225\"   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=9+x%5E2+%2B+54x+%2B+25y%5E2+%2B+50y+%2B+106=+225 ALIGN=MIDDLE  ALT=\"9+x%5E2+%2B+54x+%2B+25y%5E2+%2B+50y+%2B+106=+225\"   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=9+x%5E2+%2B+54x+%2B+25y%5E2+%2B+50y+%2B+106-225=0 ALIGN=MIDDLE  ALT=\"9+x%5E2+%2B+54x+%2B+25y%5E2+%2B+50y+%2B+106-225=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=9x%5E2+%2B+25y%5E2+%2B+54x+%2B+50y+-+119+=+0 ALIGN=MIDDLE  ALT=\"9x%5E2+%2B+25y%5E2+%2B+54x+%2B+50y+-+119+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >=> option d. \r<BR>\n" );
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document.write( "d. (9x)2 + (25y)2 + 54x + 50y  -  119 = 0 -> should be  9x^2 + 25y^2 + 54x + 50y  -  119 = 0 \r<BR>\n" );
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document.write( " </SPAN>\n" );
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