document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=971232>Question 971232</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Discuss the conic 4x2-y2+8x+2y-1=0<BR>\n" );
document.write( "find its foci, vertices and asymptotes if any </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #593805 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=593805\">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=4x%5E2-y%5E2%2B8x%2B2y-1=0 ALIGN=MIDDLE  ALT=\"4x%5E2-y%5E2%2B8x%2B2y-1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.......group <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >'s and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >'s\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284x%5E2%2B8x%29-%28y%5E2-2y-1%29=0 ALIGN=MIDDLE  ALT=\"%284x%5E2%2B8x%29-%28y%5E2-2y-1%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.........complete squares\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x%5E2%2B2x%29-%28y-1%29%5E2=0 ALIGN=MIDDLE  ALT=\"4%28x%5E2%2B2x%29-%28y-1%29%5E2=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%2B2x%2B1%5E2%29-4%2A1%5E2-%28y-1%29%5E2=0 ALIGN=MIDDLE  ALT=\"4%28x%5E2%2B2x%2B1%5E2%29-4%2A1%5E2-%28y-1%29%5E2=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%2B1%29%5E2-4-%28y-1%29%5E2=0 ALIGN=MIDDLE  ALT=\"4%28x%2B1%29%5E2-4-%28y-1%29%5E2=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%2B1%29%5E2-%28y-1%29%5E2=4 ALIGN=MIDDLE  ALT=\"4%28x%2B1%29%5E2-%28y-1%29%5E2=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >..........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" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x%2B1%29%5E2%2F4-%28y-1%29%5E2%2F4=4%2F4 ALIGN=MIDDLE  ALT=\"4%28x%2B1%29%5E2%2F4-%28y-1%29%5E2%2F4=4%2F4\"   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%2B1%29%5E2-+%28y-1%29%5E2%2F4+=+1 ALIGN=MIDDLE  ALT=\"%28x%2B1%29%5E2-+%28y-1%29%5E2%2F4+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ..........you have a hyperbola\r<BR>\n" );
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document.write( "compare it to the standard formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2Fa%5E2-%28y-k%29%5E2%2Fb%5E2+=+1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2Fa%5E2-%28y-k%29%5E2%2Fb%5E2+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "as you can see, <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;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <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( "then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=sqrt%281%5E2%2B2%5E2%29=sqrt%285%29 ALIGN=MIDDLE  ALT=\"c=sqrt%281%5E2%2B2%5E2%29=sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "so, <BR>\n" );
document.write( "the center is 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( "foci:<BR>\n" );
document.write( "((<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h-c ALIGN=MIDDLE  ALT=\"h-c\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <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=h%2Bc ALIGN=MIDDLE  ALT=\"h%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<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( "((<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1-sqrt%285%29 ALIGN=MIDDLE  ALT=\"-1-sqrt%285%29\"   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 (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Bsqrt%285%29 ALIGN=MIDDLE  ALT=\"-1%2Bsqrt%285%29\"   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( "or, approximately ((<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3.24 ALIGN=MIDDLE  ALT=\"-3.24\"   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  (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.24 ALIGN=MIDDLE  ALT=\"1.24\"   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( "vertices: | (<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;\" >)  and  (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   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( "semi-major axis length: <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( "semi-minor axis length: <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" );
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document.write( "asymptotes: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+2x%2B3 ALIGN=MIDDLE  ALT=\"y+=+2x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" >  and  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-2x-1 ALIGN=MIDDLE  ALT=\"y+=+-2x-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=drawing%28600%2C+600%2C+-5%2C5%2C+-10%2C+10%2C+%0D%0Acircle%28-3.24%2C1%2C.05%29%2Clocate%28-3.24%2C1%2Cf%28-3.24%2C1%29%29%2C%0D%0Acircle%281.24%2C1%2C.05%29%2Clocate%281.24%2C1%2Cf%281.24%2C1%29%29%2C%0D%0Acircle%28-1%2C1%2C.05%29%2Clocate%28-1%2C1.5%2CC%28-1%2C1%29%29%2C%0D%0Acircle%28-2%2C1.2%2C.05%29%2Clocate%28-2%2C1.2%2CV%28-2%2C1%29%29%2C%0D%0Acircle%280%2C1%2C.05%29%2Clocate%280%2C1%2CV%280%2C1%29%29%2C%0D%0A%0D%0A+graph%28600%2C+600%2C+-5%2C5%2C+-10%2C+10%2C-2x-1+%2Csqrt%28%28%28x%2B1%29%5E2-+1%294%29%2B1%2C2x%2B3%2C-sqrt%28%28%28x%2B1%29%5E2-+1%294%29%2B1%29+%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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