document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=214271>Question 214271</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the equation of each circle described.\r<BR>\n" );
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document.write( "Passes through the origin and is concentric with the circle x^2-6x+y^2-4y+4=0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #161843 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=161843\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "The equation of a circle centered at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(h,k\right)\"> with radius <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large r\"> is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (x - h)^2 + (y - k)^2 = r^2\">\r<BR>\n" );
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document.write( "Let's begin by determining the center of the given circle, which is the center of the circle we want because they are concentric.\r<BR>\n" );
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document.write( "Take the equation of the given concentric circle:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x^2-6x+y^2-4y+4=0\">\r<BR>\n" );
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document.write( "and complete both squares.  First put the constant term on the right.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x^2-6x+y^2-4y=-4\">\r<BR>\n" );
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document.write( "Remember for a squared binomial, the third term of the resulting trinomial is one-half of the second term coefficient squared.  -6 divided by 2 is -3; -3 squared is 9, so add 9 to both sides.  And -4 divided by 2 is -2; -2 squared is 4, so add 4 to both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x^2-6x+9+y^2-4y+4=-4+9+4\">\r<BR>\n" );
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document.write( "Now factor the two trinomials:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (x - 3)^2 + (y - 2)^2 = 9\">\r<BR>\n" );
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document.write( "Hence, the the given circle is centered at (3, 2) with radius 3.\r<BR>\n" );
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document.write( "Since the desired circle passes through the origin, the line segment with endpoints at the origin and at (3, 2) is a radius of the desired circle.  We need to know the measure of the radius which is the distance from (0, 0) to (3, 2).  Use the distance formula:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ d = sqrt{(3 - 0)^2 + (2 - 0)^2} = sqrt{9 + 4} = sqrt{13}\">\r<BR>\n" );
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document.write( "Now that we know the center and radius of the desired circle we can write the equation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (x - h)^2 + (y - k)^2 = r^2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (x - 3)^2 + (y - 2)^2 = \sqrt{13}^2 = 13\">\r<BR>\n" );
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document.write( "Expand the binomials:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x^2 - 6x + 9 + y^2 - 4x + 4 = 13\">\r<BR>\n" );
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document.write( "Simplify:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x^2 - 6x + y^2 - 4x = 0\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28%0D%0A500%2C+500%2C+-2%2C+8%2C+-2%2C+8%2C%0D%0Agrid%281%29%2C%0D%0Acircle%283%2C2%2C3%29%2C%0D%0Ared%28circle%283%2C2%2Csqrt%2813%29%29%29%2C%0D%0Ablue%28circle%283%2C2%2C.1%29%29%2C%0D%0Ablue%28line%280%2C0%2C3%2C2%29%29%2C%0D%0Ared%28locate%283.05%2C1.9%2CC%283%2C2%29%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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