document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1076714>Question 1076714</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find an equation of the circle that passes through the points (2,3), (4,5), and (0,-3). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #691338 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=691338\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  Find an equation of the circle that passes through the points (<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;\" >), (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   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=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >). \r<BR>\n" );
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document.write( "The standard form equation of a circle is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28x-h%29%5E2%2B%28y-k%29%5E2=r%5E2 ALIGN=MIDDLE  ALT=\"+%28x-h%29%5E2%2B%28y-k%29%5E2=r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" > are the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > 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;\" > coordinates of the center of the circle \r<BR>\n" );
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document.write( "first find  coordinates of the center  (<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;\" >) using given points\r<BR>\n" );
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document.write( "The first step is to set up these 3 equations by plugging the x- and y-coordinates of the points <BR>\n" );
document.write( "(<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;\" >), (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   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=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >)  into the circle formula:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282+-+h%29%5E2+%2B+%283+-+k%29%5E2+=+r%5E2 ALIGN=MIDDLE  ALT=\"%282+-+h%29%5E2+%2B+%283+-+k%29%5E2+=+r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.................eq1<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284+-+h%29%5E2+%2B+%285+-+k%29%5E2+=+r%5E2 ALIGN=MIDDLE  ALT=\"%284+-+h%29%5E2+%2B+%285+-+k%29%5E2+=+r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >..................eq2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%280+-+h%29%5E2+%2B+%28-3+-+k%29%5E2+=+r%5E2 ALIGN=MIDDLE  ALT=\"%280+-+h%29%5E2+%2B+%28-3+-+k%29%5E2+=+r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.................eq3<BR>\n" );
document.write( "----------------------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+-+4+h+%2B+k%5E2+-+6+k+%2B+13+=+r%5E2 ALIGN=MIDDLE  ALT=\"h%5E2+-+4+h+%2B+k%5E2+-+6+k+%2B+13+=+r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.................eq1<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+-+8+h+%2B+k%5E2+-+10+k+%2B+41=+r%5E2 ALIGN=MIDDLE  ALT=\"h%5E2+-+8+h+%2B+k%5E2+-+10+k+%2B+41=+r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >..................eq2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+%2B+k%5E2+%2B+6+k+%2B+9+=+r%5E2 ALIGN=MIDDLE  ALT=\"h%5E2+%2B+k%5E2+%2B+6+k+%2B+9+=+r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >...............eq3\r<BR>\n" );
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document.write( "--------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+-+4+h+%2B+k%5E2+-+6+k+%2B+13+=+r%5E2 ALIGN=MIDDLE  ALT=\"h%5E2+-+4+h+%2B+k%5E2+-+6+k+%2B+13+=+r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.................eq1<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+-+8+h+%2B+k%5E2+-+10+k+%2B+41=+r%5E2 ALIGN=MIDDLE  ALT=\"h%5E2+-+8+h+%2B+k%5E2+-+10+k+%2B+41=+r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >..................eq2...............subtract eq2 from eq1<BR>\n" );
document.write( "---------------------------------------------------------<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+-+4+h+%2B+k%5E2+-+6+k+%2B+13-%28h%5E2+-+8+h+%2B+k%5E2+-+10+k+%2B+41%29=+r%5E2-r%5E2 ALIGN=MIDDLE  ALT=\"h%5E2+-+4+h+%2B+k%5E2+-+6+k+%2B+13-%28h%5E2+-+8+h+%2B+k%5E2+-+10+k+%2B+41%29=+r%5E2-r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+-+4+h+%2B+k%5E2+-+6+k+%2B+13-h%5E2+%2B8+h-+k%5E2+%2B10+k+-+41=+0 ALIGN=MIDDLE  ALT=\"h%5E2+-+4+h+%2B+k%5E2+-+6+k+%2B+13-h%5E2+%2B8+h-+k%5E2+%2B10+k+-+41=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4+h+%2B+4+k+-+28=+0 ALIGN=MIDDLE  ALT=\"4+h+%2B+4+k+-+28=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >............simplify<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+h+%2B++k+-+7=+0 ALIGN=MIDDLE  ALT=\"+h+%2B++k+-+7=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.............eq1a\r<BR>\n" );
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document.write( "h^2 - 4h + k^2 - 6 k + 13 = r^2.................eq1<BR>\n" );
document.write( "h^2 + k^2 + 6k + 9 = r^2...............eq3...........subtract eq3 from eq1<BR>\n" );
document.write( "------------------------------------------<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+-+4h+%2B+k%5E2+-+6k+%2B+13++-%28h%5E2+%2B+k%5E2+%2B+6+k+%2B+9+%29=+r%5E2-r%5E2 ALIGN=MIDDLE  ALT=\"h%5E2+-+4h+%2B+k%5E2+-+6k+%2B+13++-%28h%5E2+%2B+k%5E2+%2B+6+k+%2B+9+%29=+r%5E2-r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+-+4h+%2B+k%5E2+-+6k+%2B+13++-h%5E2+-+k%5E2+-+6+k+-+9+=0 ALIGN=MIDDLE  ALT=\"h%5E2+-+4h+%2B+k%5E2+-+6k+%2B+13++-h%5E2+-+k%5E2+-+6+k+-+9+=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4+h+-+12k+%2B+4+=0 ALIGN=MIDDLE  ALT=\"-4+h+-+12k+%2B+4+=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >............divide by -4<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+h+%2B3k+-+1+=0 ALIGN=MIDDLE  ALT=\"+h+%2B3k+-+1+=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >..............................eq2a\r<BR>\n" );
document.write( "\n" );
document.write( "use <BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+%2B++k+-+7=+0 ALIGN=MIDDLE  ALT=\"h+%2B++k+-+7=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >...............eq1a<BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+h+%2B3+k+-+1+=0 ALIGN=MIDDLE  ALT=\"+h+%2B3+k+-+1+=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >..............................eq2a<BR>\n" );
document.write( "-------------------------------------------------------------subtract eq1a from eq2a\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+h+%2B3+k+-+1+-%28h+%2B++k+-+7%29=0 ALIGN=MIDDLE  ALT=\"+h+%2B3+k+-+1+-%28h+%2B++k+-+7%29=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=h+%2B3+k+-+1+-h+-++k+%2B+7=0 ALIGN=MIDDLE  ALT=\"h+%2B3+k+-+1+-h+-++k+%2B+7=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=2+k++%2B+6=0 ALIGN=MIDDLE  ALT=\"2+k++%2B+6=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=2+k++=-6 ALIGN=MIDDLE  ALT=\"2+k++=-6\"   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=+highlight%28k++=+-3%29 ALIGN=MIDDLE  ALT=\"+highlight%28k++=+-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "now find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   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+%2B++k+-+7=+0 ALIGN=MIDDLE  ALT=\"h+%2B++k+-+7=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >...............eq1a\r<BR>\n" );
document.write( "\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+-3-+7=+0 ALIGN=MIDDLE  ALT=\"h+-3-+7=+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=h-10=0 ALIGN=MIDDLE  ALT=\"h-10=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=h=10 ALIGN=MIDDLE  ALT=\"h=10\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "so, center is at  (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   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" );
document.write( "\n" );
document.write( "use one point, h and k to find r\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282+-+10%29%5E2+%2B+%283+-+%28-3%29%29%5E2+=+r%5E2 ALIGN=MIDDLE  ALT=\"%282+-+10%29%5E2+%2B+%283+-+%28-3%29%29%5E2+=+r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.................eq1\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28+-+8%29%5E2+%2B+%283+%2B3%29%5E2+=+r%5E2 ALIGN=MIDDLE  ALT=\"%28+-+8%29%5E2+%2B+%283+%2B3%29%5E2+=+r%5E2\"   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=64+%2B+36+=+r%5E2 ALIGN=MIDDLE  ALT=\"64+%2B+36+=+r%5E2\"   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=r%5E2=100 ALIGN=MIDDLE  ALT=\"r%5E2=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=r=10 ALIGN=MIDDLE  ALT=\"r=10\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "and, 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-10%29%5E2%2B%28y%2B3%29%5E2=10%5E2 ALIGN=MIDDLE  ALT=\"%28x-10%29%5E2%2B%28y%2B3%29%5E2=10%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "(2,3), (4,5), and (0,-3).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+600%2C+600%2C+-20%2C+20%2C+-20%2C+20%2C++%0D%0Acircle%2810%2C-3%2C.13%29%2Clocate%2810%2C-2.8%2CC%2810%2C-3%29%29%2C%0D%0Acircle%282%2C3%2C.13%29%2Clocate%282%2C2.8%2Cp%282%2C3%29%29%2C%0D%0Acircle%284%2C5%2C.13%29%2Clocate%284%2C4.8%2Cp%284%2C5%29%29%2C%0D%0Acircle%280%2C-3%2C.13%29%2Clocate%280%2C-3%2Cp%280%2C-3%29%29%2C%0D%0Acircle%2810%2C-3%2C10%29%2C%0D%0Agraph%28+600%2C+600%2C+-20%2C+20%2C+-20%2C+20%2C+0%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );