document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1180259>Question 1180259</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Identify the equations of ellipses whose major axis lengths are twice their minor axis lengths.\r<BR>\n" );
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document.write( "a. 4x^2+25y^2+32x-250y+589=0<BR>\n" );
document.write( "b. 2x^2+8y^2-12x+16y-174=0<BR>\n" );
document.write( "c. 4x^2+y^2+16x+4y+4=0<BR>\n" );
document.write( "d. 3x^2+12y^2+18x-24y-69=0<BR>\n" );
document.write( "e. 16x^2+y^2-64x+8y+16=0<BR>\n" );
document.write( "f. x^2+9y^2-2x+18y-71=0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #809910 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=809910\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Identify the equations of ellipses whose major axis lengths are twice their minor axis lengths.\r<BR>\n" );
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document.write( "recall: recall: The equation of an ellipse is<BR>\n" );
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;\" >-> horizontal major axis, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3Eb ALIGN=MIDDLE  ALT=\"a%3Eb\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2Fb%5E2%2B%28y-k%29%5E2%2Fa%5E2=1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2Fb%5E2%2B%28y-k%29%5E2%2Fa%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >-> vertical major axis, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3Eb ALIGN=MIDDLE  ALT=\"a%3Eb\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "The center is (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and the larger of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+ ALIGN=MIDDLE  ALT=\"a+\"   DISABLED_style= \"max-width:90%; height:auto;\" >the major axis ,<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 minor axis.\r<BR>\n" );
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document.write( "a. <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2%2B25y%5E2%2B32x-250y%2B589=0 ALIGN=MIDDLE  ALT=\"4x%5E2%2B25y%5E2%2B32x-250y%2B589=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >......competing squares you will get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284x%5E2%2B32x%29%2B%2825y%5E2-250y%29%2B589=0 ALIGN=MIDDLE  ALT=\"%284x%5E2%2B32x%29%2B%2825y%5E2-250y%29%2B589=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=4%28x%5E2%2B8x%29%2B25%28y%5E2-10y%29%2B589=0 ALIGN=MIDDLE  ALT=\"4%28x%5E2%2B8x%29%2B25%28y%5E2-10y%29%2B589=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=4%28x%5E2%2B8x%2Bb%5E2%29-4b%5E2%2B25%28y%5E2-10y%2Bb%5E2%29-25b%5E2%2B589=0 ALIGN=MIDDLE  ALT=\"4%28x%5E2%2B8x%2Bb%5E2%29-4b%5E2%2B25%28y%5E2-10y%2Bb%5E2%29-25b%5E2%2B589=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=4%28x%5E2%2B8x%2B4%5E2%29-4%2A4%5E2%2B25%28y%5E2-10y%2B5%5E2%29-25%2A5%5E2%2B589=0 ALIGN=MIDDLE  ALT=\"4%28x%5E2%2B8x%2B4%5E2%29-4%2A4%5E2%2B25%28y%5E2-10y%2B5%5E2%29-25%2A5%5E2%2B589=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=4%28x%2B4%29%5E2-64%2B25%28y-5%29%5E2-625%2B589=0 ALIGN=MIDDLE  ALT=\"4%28x%2B4%29%5E2-64%2B25%28y-5%29%5E2-625%2B589=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=4%28x%2B4%29%5E2%2B25%28y-5%29%5E2-100=0 ALIGN=MIDDLE  ALT=\"4%28x%2B4%29%5E2%2B25%28y-5%29%5E2-100=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=4%28x%2B4%29%5E2%2B25%28y-5%29%5E2=100 ALIGN=MIDDLE  ALT=\"4%28x%2B4%29%5E2%2B25%28y-5%29%5E2=100\"   DISABLED_style= \"max-width:90%; height:auto;\" >........both sides divide by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100 ALIGN=MIDDLE  ALT=\"100\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x%2B4%29%5E2%2F100%2B25%28y-5%29%5E2%2F100=100%2F100 ALIGN=MIDDLE  ALT=\"4%28x%2B4%29%5E2%2F100%2B25%28y-5%29%5E2%2F100=100%2F100\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B4%29%5E2%2F25%2B%28y-5%29%5E2%2F4=1 ALIGN=MIDDLE  ALT=\"%28x%2B4%29%5E2%2F25%2B%28y-5%29%5E2%2F4=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B4%29%5E2%2F5%5E2%2B%28y-5%29%5E2%2F2%5E2=1 ALIGN=MIDDLE  ALT=\"%28x%2B4%29%5E2%2F5%5E2%2B%28y-5%29%5E2%2F2%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-4 ALIGN=MIDDLE  ALT=\"h=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=5 ALIGN=MIDDLE  ALT=\"k=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "major axis:<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;\" ><BR>\n" );
document.write( "minor axis:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=2 ALIGN=MIDDLE  ALT=\"b=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "=> major axis length is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=not ALIGN=MIDDLE  ALT=\"not\"   DISABLED_style= \"max-width:90%; height:auto;\" > twice the  length of minor axis \r<BR>\n" );
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document.write( "<BR>\n" );
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document.write( "b. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2B8y%5E2-12x%2B16y-174=0 ALIGN=MIDDLE  ALT=\"2x%5E2%2B8y%5E2-12x%2B16y-174=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%5E2-12x%29%2B%288y%5E2%2B16y%29=174 ALIGN=MIDDLE  ALT=\"%282x%5E2-12x%29%2B%288y%5E2%2B16y%29=174\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28x%5E2-6x%2Bb%5E2%29-2b%5E2%2B8%28y%5E2%2B2y%2Bb%5E2%29-8b%5E2=174 ALIGN=MIDDLE  ALT=\"2%28x%5E2-6x%2Bb%5E2%29-2b%5E2%2B8%28y%5E2%2B2y%2Bb%5E2%29-8b%5E2=174\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28x%5E2-6x%2B3%5E2%29-2%2A3%5E2%2B8%28y%5E2%2B2y%2B1%5E2%29-8%2A1%5E2=174 ALIGN=MIDDLE  ALT=\"2%28x%5E2-6x%2B3%5E2%29-2%2A3%5E2%2B8%28y%5E2%2B2y%2B1%5E2%29-8%2A1%5E2=174\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28x-3%29%5E2-18%2B8%28y%2B1%29%5E2-8=174 ALIGN=MIDDLE  ALT=\"2%28x-3%29%5E2-18%2B8%28y%2B1%29%5E2-8=174\"   DISABLED_style= \"max-width:90%; height:auto;\" >....simplify, both sides divide by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-3%29%5E2-9%2B4%28y%2B1%29%5E2-4=87 ALIGN=MIDDLE  ALT=\"%28x-3%29%5E2-9%2B4%28y%2B1%29%5E2-4=87\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-3%29%5E2%2B4%28y%2B1%29%5E2=87%2B9%2B4 ALIGN=MIDDLE  ALT=\"%28x-3%29%5E2%2B4%28y%2B1%29%5E2=87%2B9%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-3%29%5E2%2B4%28y%2B1%29%5E2=100 ALIGN=MIDDLE  ALT=\"%28x-3%29%5E2%2B4%28y%2B1%29%5E2=100\"   DISABLED_style= \"max-width:90%; height:auto;\" >........both sides divide by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100 ALIGN=MIDDLE  ALT=\"100\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-3%29%5E2%2F100%2B4%28y%2B1%29%5E2%2F100=100%2F100 ALIGN=MIDDLE  ALT=\"%28x-3%29%5E2%2F100%2B4%28y%2B1%29%5E2%2F100=100%2F100\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-3%29%5E2%2F100%2B%28y%2B1%29%5E2%2F25=1 ALIGN=MIDDLE  ALT=\"%28x-3%29%5E2%2F100%2B%28y%2B1%29%5E2%2F25=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-3%29%5E2%2F10%5E2%2B%28y%2B1%29%5E2%2F5%5E2=1 ALIGN=MIDDLE  ALT=\"%28x-3%29%5E2%2F10%5E2%2B%28y%2B1%29%5E2%2F5%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<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;\" ><BR>\n" );
document.write( "major axis:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=10 ALIGN=MIDDLE  ALT=\"a+=10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "minor axis:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=5 ALIGN=MIDDLE  ALT=\"b=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "=> major axis length  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=is ALIGN=MIDDLE  ALT=\"is\"   DISABLED_style= \"max-width:90%; height:auto;\" > twice the  length of minor axis \r<BR>\n" );
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document.write( "c. <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2%2By%5E2%2B16x%2B4y%2B4=0 ALIGN=MIDDLE  ALT=\"4x%5E2%2By%5E2%2B16x%2B4y%2B4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284x%5E2%2B16x%29%2B%28y%5E2%2B4y%29=-4 ALIGN=MIDDLE  ALT=\"%284x%5E2%2B16x%29%2B%28y%5E2%2B4y%29=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x%5E2%2B4x%2Bb%5E2%29-4b%5E2%2B%28y%5E2%2B4y%2Bb%5E2%29-b%5E2=-4 ALIGN=MIDDLE  ALT=\"4%28x%5E2%2B4x%2Bb%5E2%29-4b%5E2%2B%28y%5E2%2B4y%2Bb%5E2%29-b%5E2=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x%5E2%2B4x%2B2%5E2%29-4%2A2%5E2%2B%28y%5E2%2B4y%2B2%5E2%29-2%5E2=-4 ALIGN=MIDDLE  ALT=\"4%28x%5E2%2B4x%2B2%5E2%29-4%2A2%5E2%2B%28y%5E2%2B4y%2B2%5E2%29-2%5E2=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x%2B2%29%5E2-16%2B%28y%2B2%29%5E2-4=-4 ALIGN=MIDDLE  ALT=\"4%28x%2B2%29%5E2-16%2B%28y%2B2%29%5E2-4=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x%2B2%29%5E2%2B%28y%2B2%29%5E2=-4%2B16%2B4 ALIGN=MIDDLE  ALT=\"4%28x%2B2%29%5E2%2B%28y%2B2%29%5E2=-4%2B16%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x%2B2%29%5E2%2B%28y%2B2%29%5E2=16 ALIGN=MIDDLE  ALT=\"4%28x%2B2%29%5E2%2B%28y%2B2%29%5E2=16\"   DISABLED_style= \"max-width:90%; height:auto;\" >......both sides divide by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16 ALIGN=MIDDLE  ALT=\"16\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x%2B2%29%5E2%2F16%2B%28y%2B2%29%5E2%2F16=16%2F16 ALIGN=MIDDLE  ALT=\"4%28x%2B2%29%5E2%2F16%2B%28y%2B2%29%5E2%2F16=16%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B2%29%5E2%2F4%2B%28y%2B2%29%5E2%2F16=1 ALIGN=MIDDLE  ALT=\"%28x%2B2%29%5E2%2F4%2B%28y%2B2%29%5E2%2F16=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B2%29%5E2%2F2%5E2%2B%28y%2B2%29%5E2%2F4%5E2=1 ALIGN=MIDDLE  ALT=\"%28x%2B2%29%5E2%2F2%5E2%2B%28y%2B2%29%5E2%2F4%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=-2 ALIGN=MIDDLE  ALT=\"h=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-2 ALIGN=MIDDLE  ALT=\"k=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "major axis:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=4 ALIGN=MIDDLE  ALT=\"b+=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "minor axis:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=2 ALIGN=MIDDLE  ALT=\"a=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "=> major axis length  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=is ALIGN=MIDDLE  ALT=\"is\"   DISABLED_style= \"max-width:90%; height:auto;\" > twice the  length of minor axis\r<BR>\n" );
document.write( "\n" );
document.write( "d. \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2B12y%5E2%2B18x-24y-69=0 ALIGN=MIDDLE  ALT=\"3x%5E2%2B12y%5E2%2B18x-24y-69=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >...simplify, both sides divide by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B4y%5E2%2B6x-8y-23=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B4y%5E2%2B6x-8y-23=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2B6x%29%2B%284y%5E2-8y%29=23 ALIGN=MIDDLE  ALT=\"%28x%5E2%2B6x%29%2B%284y%5E2-8y%29=23\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2B6x%2Bb%5E2%29-b%5E2%2B4%28y%5E2-2y%2Bb%5E2%29-4b%5E2=23 ALIGN=MIDDLE  ALT=\"%28x%5E2%2B6x%2Bb%5E2%29-b%5E2%2B4%28y%5E2-2y%2Bb%5E2%29-4b%5E2=23\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2B6x%2B3%5E2%29-3%5E2%2B4%28y%5E2-2y%2B1%5E2%29-4%2A1%5E2=23 ALIGN=MIDDLE  ALT=\"%28x%5E2%2B6x%2B3%5E2%29-3%5E2%2B4%28y%5E2-2y%2B1%5E2%29-4%2A1%5E2=23\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%5E2-9%2B4%28y-1%29%5E2-4=23 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%5E2-9%2B4%28y-1%29%5E2-4=23\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%5E2%2B4%28y-1%29%5E2=23%2B9%2B4 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%5E2%2B4%28y-1%29%5E2=23%2B9%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%5E2%2B4%28y-1%29%5E2=36 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%5E2%2B4%28y-1%29%5E2=36\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%5E2%2F36%2B4%28y-1%29%5E2%2F36=36%2F36 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%5E2%2F36%2B4%28y-1%29%5E2%2F36=36%2F36\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%5E2%2F36%2B%28y-1%29%5E2%2F9=1 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%5E2%2F36%2B%28y-1%29%5E2%2F9=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%5E2%2F6%5E2%2B%28y-1%29%5E2%2F3%5E2=1 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%5E2%2F6%5E2%2B%28y-1%29%5E2%2F3%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<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;\" ><BR>\n" );
document.write( "major axis:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=6 ALIGN=MIDDLE  ALT=\"a+=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "minor axis:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=3 ALIGN=MIDDLE  ALT=\"b=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "=> major axis length  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=is ALIGN=MIDDLE  ALT=\"is\"   DISABLED_style= \"max-width:90%; height:auto;\" > twice the  length of minor axis\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "e. \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16x%5E2%2By%5E2-64x%2B8y%2B16=0 ALIGN=MIDDLE  ALT=\"16x%5E2%2By%5E2-64x%2B8y%2B16=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2816x%5E2-64x%29%2B%28y%5E2%2B8y%29=-16 ALIGN=MIDDLE  ALT=\"%2816x%5E2-64x%29%2B%28y%5E2%2B8y%29=-16\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16%28x%5E2-4x%2Bb%5E2%29-16b%5E2%2B%28y%5E2%2B8y%2Bb%5E2%29-b%5E2=-16 ALIGN=MIDDLE  ALT=\"16%28x%5E2-4x%2Bb%5E2%29-16b%5E2%2B%28y%5E2%2B8y%2Bb%5E2%29-b%5E2=-16\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16%28x%5E2-4x%2B2%5E2%29-16%2A2%5E2%2B%28y%5E2%2B8y%2B4%5E2%29-4%5E2=-16 ALIGN=MIDDLE  ALT=\"16%28x%5E2-4x%2B2%5E2%29-16%2A2%5E2%2B%28y%5E2%2B8y%2B4%5E2%29-4%5E2=-16\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16%28x-2%29%5E2-64%2B%28y%2B4%29%5E2-16=-16 ALIGN=MIDDLE  ALT=\"16%28x-2%29%5E2-64%2B%28y%2B4%29%5E2-16=-16\"   DISABLED_style= \"max-width:90%; height:auto;\" >...simplify, both sides divide by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16 ALIGN=MIDDLE  ALT=\"16\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%5E2-4%2B%28y%2B4%29%5E2-1=-1 ALIGN=MIDDLE  ALT=\"%28x-2%29%5E2-4%2B%28y%2B4%29%5E2-1=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%5E2%2B%28y%2B4%29%5E2=-1%2B4%2B1 ALIGN=MIDDLE  ALT=\"%28x-2%29%5E2%2B%28y%2B4%29%5E2=-1%2B4%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%5E2%2B%28y%2B4%29%5E2=4 ALIGN=MIDDLE  ALT=\"%28x-2%29%5E2%2B%28y%2B4%29%5E2=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.........both sides divide by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%5E2%2F4%2B%28y%2B4%29%5E2%2F4=1 ALIGN=MIDDLE  ALT=\"%28x-2%29%5E2%2F4%2B%28y%2B4%29%5E2%2F4=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%5E2%2F2%5E2%2B%28y%2B4%29%5E2%2F2%5E2=1 ALIGN=MIDDLE  ALT=\"%28x-2%29%5E2%2F2%5E2%2B%28y%2B4%29%5E2%2F2%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=2 ALIGN=MIDDLE  ALT=\"h=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-4 ALIGN=MIDDLE  ALT=\"k=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "major axis:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=2 ALIGN=MIDDLE  ALT=\"a+=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "minor axis:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=2 ALIGN=MIDDLE  ALT=\"b=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "=> major axis length  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=is ALIGN=MIDDLE  ALT=\"is\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=not ALIGN=MIDDLE  ALT=\"not\"   DISABLED_style= \"max-width:90%; height:auto;\" > twice the  length of minor axis\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "f. \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B9y%5E2-2x%2B18y-71=0 ALIGN=MIDDLE  ALT=\"x%5E2%2B9y%5E2-2x%2B18y-71=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2-2x%29%2B%289y%5E2%2B18y%29=71 ALIGN=MIDDLE  ALT=\"%28x%5E2-2x%29%2B%289y%5E2%2B18y%29=71\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2-2x%2Bb%5E2%29-b%5E2%2B9%28y%5E2%2B2y%2Bb%5E2%29-9b%5E2=71 ALIGN=MIDDLE  ALT=\"%28x%5E2-2x%2Bb%5E2%29-b%5E2%2B9%28y%5E2%2B2y%2Bb%5E2%29-9b%5E2=71\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2-2x%2B1%5E2%29-1%5E2%2B9%28y%5E2%2B2y%2B1%5E2%29-9%2A1%5E2=71 ALIGN=MIDDLE  ALT=\"%28x%5E2-2x%2B1%5E2%29-1%5E2%2B9%28y%5E2%2B2y%2B1%5E2%29-9%2A1%5E2=71\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%5E2-1%2B9%28y%2B1%29%5E2-9=71 ALIGN=MIDDLE  ALT=\"%28x-1%29%5E2-1%2B9%28y%2B1%29%5E2-9=71\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%5E2%2B9%28y%2B1%29%5E2=71%2B10 ALIGN=MIDDLE  ALT=\"%28x-1%29%5E2%2B9%28y%2B1%29%5E2=71%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%5E2%2B9%28y%2B1%29%5E2=81 ALIGN=MIDDLE  ALT=\"%28x-1%29%5E2%2B9%28y%2B1%29%5E2=81\"   DISABLED_style= \"max-width:90%; height:auto;\" >.......both sides divide by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=81 ALIGN=MIDDLE  ALT=\"81\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%5E2%2F81%2B9%28y%2B1%29%5E2%2F81=1 ALIGN=MIDDLE  ALT=\"%28x-1%29%5E2%2F81%2B9%28y%2B1%29%5E2%2F81=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%5E2%2F81%2B%28y%2B1%29%5E2%2F9=1 ALIGN=MIDDLE  ALT=\"%28x-1%29%5E2%2F81%2B%28y%2B1%29%5E2%2F9=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%5E2%2F9%5E2%2B%28y%2B1%29%5E2%2F3%5E2=1 ALIGN=MIDDLE  ALT=\"%28x-1%29%5E2%2F9%5E2%2B%28y%2B1%29%5E2%2F3%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=1 ALIGN=MIDDLE  ALT=\"h=1\"   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;\" ><BR>\n" );
document.write( "major axis:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=9 ALIGN=MIDDLE  ALT=\"a+=9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "minor axis:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=3 ALIGN=MIDDLE  ALT=\"b=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "=> major axis length  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=is ALIGN=MIDDLE  ALT=\"is\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=not ALIGN=MIDDLE  ALT=\"not\"   DISABLED_style= \"max-width:90%; height:auto;\" > twice the  length of minor axis\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );