document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=771638>Question 771638</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the equation of the circle in the form X^2+y^2+Dx+Ey+F=0 of the circle through points (1,1) (2,-1), (2,3)<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 #470347 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=470347\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Consider the general equation of a circle as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X%5E2+%2B+Y%5E2+%2B+DX+%2B+EY+%2B+F+=+0 ALIGN=MIDDLE  ALT=\"X%5E2+%2B+Y%5E2+%2B+DX+%2B+EY+%2B+F+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "given: points (<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;\" >) (<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;\" >), (<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=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "Using the given three points we derive our equation :\r<BR>\n" );
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document.write( "From the point (<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;\" >) we get the equation \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B+1+%2B+D+%2B+E+%2B+F+=+0 ALIGN=MIDDLE  ALT=\"1%2B+1+%2B+D+%2B+E+%2B+F+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D+%2B+E+%2B+F%2B+2+=+0 ALIGN=MIDDLE  ALT=\"D+%2B+E+%2B+F%2B+2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ........  equation 1\r<BR>\n" );
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document.write( "From the point (<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;\" >) we get \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4+%2B+1+%2B+2D+-E+%2B+F+=+0 ALIGN=MIDDLE  ALT=\"4+%2B+1+%2B+2D+-E+%2B+F+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2D+-E+%2B+F%2B5+=+0 ALIGN=MIDDLE  ALT=\"2D+-E+%2B+F%2B5+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >........  equation 2\r<BR>\n" );
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document.write( "From the point (<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=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >) we get \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4+%2B+9+%2B2D+%2B+3E+%2B+F+=+0 ALIGN=MIDDLE  ALT=\"4+%2B+9+%2B2D+%2B+3E+%2B+F+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2D+%2B+3E+%2B+F%2B13+=+0 ALIGN=MIDDLE  ALT=\"2D+%2B+3E+%2B+F%2B13+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.......  equation 3\r<BR>\n" );
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document.write( "We now have a system of three equations :\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D+%2B+E+%2B+F%2B+2+=+0 ALIGN=MIDDLE  ALT=\"D+%2B+E+%2B+F%2B+2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ........  equation 1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2D+-E+%2B+F%2B5+=+0 ALIGN=MIDDLE  ALT=\"2D+-E+%2B+F%2B5+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >........  equation 2\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2D+%2B+3E+%2B+F%2B13+=+0 ALIGN=MIDDLE  ALT=\"2D+%2B+3E+%2B+F%2B13+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.......  equation 3\r<BR>\n" );
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document.write( "We now use elimination method to find the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D ALIGN=MIDDLE  ALT=\"D\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+E ALIGN=MIDDLE  ALT=\"+E\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F ALIGN=MIDDLE  ALT=\"F\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Using equation 1 and eq. 2 to eliminate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F ALIGN=MIDDLE  ALT=\"F\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we do this by subtracting equation 2 from eq. 1 :\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D+%2B+E+%2B+F%2B+2-%28+2D+-E+%2B+F%2B5+%29=+0 ALIGN=MIDDLE  ALT=\"D+%2B+E+%2B+F%2B+2-%28+2D+-E+%2B+F%2B5+%29=+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=D+%2B+E+%2B+F%2B+2-2D+%2BE+-+F-5+=+0 ALIGN=MIDDLE  ALT=\"D+%2B+E+%2B+F%2B+2-2D+%2BE+-+F-5+=+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=2E+-2D-3+=+0 ALIGN=MIDDLE  ALT=\"2E+-2D-3+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >..... let this be equation 4\r<BR>\n" );
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document.write( "Using Equation 2 and eqn 3, if we subtract eq. 3 from eq. 2 we can eliminate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F ALIGN=MIDDLE  ALT=\"F\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D ALIGN=MIDDLE  ALT=\"D\"   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=2D+-E+%2B+F%2B5+-%282D+%2B+3E+%2B+F%2B13+%29=+0 ALIGN=MIDDLE  ALT=\"2D+-E+%2B+F%2B5+-%282D+%2B+3E+%2B+F%2B13+%29=+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=2D+-E+%2B+F%2B5+-2D-3E+-+F-13+=+0 ALIGN=MIDDLE  ALT=\"2D+-E+%2B+F%2B5+-2D-3E+-+F-13+=+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=-E+%2B5+-3E+-13+=+0 ALIGN=MIDDLE  ALT=\"-E+%2B5+-3E+-13+=+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=-4E++-8+=+0 ALIGN=MIDDLE  ALT=\"-4E++-8+=+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=-E++-2+=+0 ALIGN=MIDDLE  ALT=\"-E++-2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > => <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28E+=-2%29 ALIGN=MIDDLE  ALT=\"highlight%28E+=-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "go back to eq.4 and plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E ALIGN=MIDDLE  ALT=\"E\"   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=2%28-2%29+-2D-3+=+0 ALIGN=MIDDLE  ALT=\"2%28-2%29+-2D-3+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.......solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D ALIGN=MIDDLE  ALT=\"D\"   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+-3+=+2D ALIGN=MIDDLE  ALT=\"-4+-3+=+2D\"   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=-7+=+2D ALIGN=MIDDLE  ALT=\"-7+=+2D\"   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=-7+%2F2=+D ALIGN=MIDDLE  ALT=\"-7+%2F2=+D\"   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=highlight%28D=-3.5%29 ALIGN=MIDDLE  ALT=\"highlight%28D=-3.5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "go back to eq.1 and plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E ALIGN=MIDDLE  ALT=\"E\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D ALIGN=MIDDLE  ALT=\"D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F ALIGN=MIDDLE  ALT=\"F\"   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=-3.5-2+%2B+F%2B+2+=+0 ALIGN=MIDDLE  ALT=\"-3.5-2+%2B+F%2B+2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ........  equation 1\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3.5+%2B+F+=+0 ALIGN=MIDDLE  ALT=\"-3.5+%2B+F+=+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=highlight%28F+=+3.5%29 ALIGN=MIDDLE  ALT=\"highlight%28F+=+3.5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We get this general equation by substituting \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D+=+-3.5 ALIGN=MIDDLE  ALT=\"D+=+-3.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E+=+-2 ALIGN=MIDDLE  ALT=\"E+=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F+=+3.5 ALIGN=MIDDLE  ALT=\"F+=+3.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the general equation of the circle \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X%5E2+%2B+Y%5E2+%2B+DX+%2B+EY+%2B+F+=+0 ALIGN=MIDDLE  ALT=\"X%5E2+%2B+Y%5E2+%2B+DX+%2B+EY+%2B+F+=+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+%2B+Y%5E2-3.5X+-2Y+%2B+3.5+=+0 ALIGN=MIDDLE  ALT=\"X%5E2+%2B+Y%5E2-3.5X+-2Y+%2B+3.5+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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