document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=847330>Question 847330</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find an equation of the circle that has center (4,-6)  and passes through (-3,-4)  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #510352 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=510352\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The standard form of the equation for a circle 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+%2B+%28y+%2B+k%29%5E2+=+r%5E2 ALIGN=MIDDLE  ALT=\"+%28x+-+h%29%5E2+%2B+%28y+%2B+k%29%5E2+=+r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+ ALIGN=MIDDLE  ALT=\"h+\"   DISABLED_style= \"max-width:90%; height:auto;\" > is  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;\" > coordinate of the center of the circle , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > coordinate of the center of the circle, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+ ALIGN=MIDDLE  ALT=\"r+\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the radius of the circle .\r<BR>\n" );
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document.write( "the circle that has center (4,-6) and passes through (-3,-4) \r<BR>\n" );
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document.write( " given:<BR>\n" );
document.write( " the center of the circle at (<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=-+6 ALIGN=MIDDLE  ALT=\"-+6\"   DISABLED_style= \"max-width:90%; height:auto;\" >) <BR>\n" );
document.write( "one  point  the circle passes through: (<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;\" >) \r<BR>\n" );
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document.write( "The only thing we are missing to complete the equation for the circle is to know (if given) or derive (if not given), <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the radius of the circle. \r<BR>\n" );
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document.write( " With the two points given: center  (<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=-+6 ALIGN=MIDDLE  ALT=\"-+6\"   DISABLED_style= \"max-width:90%; height:auto;\" >)  and circle passes through (<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;\" >) ; if we connect those two points together, that line will be equal the radius of the circle. \r<BR>\n" );
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document.write( "To determine the radius, think about a right triangle that has as its \"<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >\" side 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;\" >\" side the rise and run difference between the two given points. And, knowing the Pythagorean theorem to determine the hypotenuse of a right triangle: \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+%2B+b%5E2+ ALIGN=MIDDLE  ALT=\"c%5E2=+a%5E2+%2B+b%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " We know that in in this instance the hypotenuse is equal to the radius of our circle, so we will substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+r%5E2=a%5E2+%2B+b%5E2+ ALIGN=MIDDLE  ALT=\"+r%5E2=a%5E2+%2B+b%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If we take the square root of both sides of the previous equation we get: <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+r+=+sqrt%28a%5E2+%2B+b%5E2%29 ALIGN=MIDDLE  ALT=\"+r+=+sqrt%28a%5E2+%2B+b%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " To solve, you need: \r<BR>\n" );
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document.write( "<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 difference in the two <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+ ALIGN=MIDDLE  ALT=\"x+\"   DISABLED_style= \"max-width:90%; height:auto;\" >coordinates (from the two points given): <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+4+-+%28-3%29+=+4%2B3=7 ALIGN=MIDDLE  ALT=\"a+=+4+-+%28-3%29+=+4%2B3=7\"   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=b+ ALIGN=MIDDLE  ALT=\"b+\"   DISABLED_style= \"max-width:90%; height:auto;\" > = the difference in the two y coordinates (from the two points given): \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=-+6+-+%28-4%29=-6%2B4+=+-2 ALIGN=MIDDLE  ALT=\"b+=-+6+-+%28-4%29=-6%2B4+=+-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=+r+=+sqrt%287%5E2+%2B+%28-2%29%5E2%29 ALIGN=MIDDLE  ALT=\"+r+=+sqrt%287%5E2+%2B+%28-2%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=+r+=+sqrt%2849%2B4+%29+ ALIGN=MIDDLE  ALT=\"+r+=+sqrt%2849%2B4+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+sqrt%2853%29 ALIGN=MIDDLE  ALT=\"r+=+sqrt%2853%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "To return to the original problem: determine the equation for the circle with the given center point and passing through the given point: \r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+-+4%29%5E2+%2B+%28y+%2B6%29%5E2+=+%28sqrt%2853%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28x+-+4%29%5E2+%2B+%28y+%2B6%29%5E2+=+%28sqrt%2853%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >   which could be simplified to\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28x+-+4%29%5E2+%2B+%28y+%2B+6%29%5E2+=53+ ALIGN=MIDDLE  ALT=\"+%28x+-+4%29%5E2+%2B+%28y+%2B+6%29%5E2+=53+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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