document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186122>Question 1186122</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the differential equation representing all circles with center on the line y = x and passing<BR>\n" );
document.write( "through the origin. Draw some representation of the family </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #817033 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=817033\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "The equation of the circle with center at (h,k) and having radius r is given by \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-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;\" >\r<BR>\n" );
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document.write( "Since the center is on the line y = x, then h = k, hence we have \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-h%29%5E2+=+r%5E2 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2+%2B+%28y-h%29%5E2+=+r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Since every circle in the family of circles passes through the point (0,0), we get \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%280-h%29%5E2+%2B+%280-h%29%5E2+=+2h%5E2+=+r%5E2 ALIGN=MIDDLE  ALT=\"%280-h%29%5E2+%2B+%280-h%29%5E2+=+2h%5E2+=+r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which implies that the equation of the family of circles 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-h%29%5E2+=+2h%5E2 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2+%2B+%28y-h%29%5E2+=+2h%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Differentiating with respect to x while treating h as a parameter,  we get \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28x-h%29+%2B+2%28y-h%29%2A%22y%27%22+=+0 ALIGN=MIDDLE  ALT=\"2%28x-h%29+%2B+2%28y-h%29%2A%22y%27%22+=+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+-+h+%2B+y%2A%22y%27%22+-+h%2A%22y%27%22+=+0 ALIGN=MIDDLE  ALT=\"x+-+h+%2B+y%2A%22y%27%22+-+h%2A%22y%27%22+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28x+%2B+y%2A%22y%27%22+=+h%281%2B%22y%27%22%29%29 ALIGN=MIDDLE  ALT=\"red%28x+%2B+y%2A%22y%27%22+=+h%281%2B%22y%27%22%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now expand the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2+%2B+%28y-h%29%5E2+=+2h%5E2 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2+%2B+%28y-h%29%5E2+=+2h%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the family of circles. This gives <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+2hx+%2B+h%5E2+%2B+y%5E2+-+2hy+%2B+h%5E2+=+2h%5E2 ALIGN=MIDDLE  ALT=\"x%5E2+-+2hx+%2B+h%5E2+%2B+y%5E2+-+2hy+%2B+h%5E2+=+2h%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which leads to \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+2hx+%2B+y%5E2+-+2hy+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-+2hx+%2B+y%5E2+-+2hy+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+y%5E2+=+2h%28x%2By%29 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+y%5E2+=+2h%28x%2By%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=%28x%5E2+%2B+y%5E2%29%2F%282%28x%2By%29%29 ALIGN=MIDDLE  ALT=\"h=%28x%5E2+%2B+y%5E2%29%2F%282%28x%2By%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Substituting for h into the equation above (in red ink), we obtain\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B+y%2A%22y%27%22+=+red%28%28%28x%5E2+%2B+y%5E2%29%2F%282%28x%2By%29%29%29%29%281%2B%22y%27%22%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22y%27%22 ALIGN=MIDDLE  ALT=\"%22y%27%22\"   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%2By%2A%22y%27%22+=+%28x%5E2+%2B+y%5E2%29%2F%282%28x%2By%29%29+%2B+%28+%28x%5E2+%2B+y%5E2%29%2F%282%28x%2By%29%29+%29%2A%22y%27%22 ALIGN=MIDDLE   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=x+-+%28x%5E2+%2B+y%5E2%29%2F%282%28x%2By%29%29+=+%28+%28x%5E2+%2B+y%5E2%29%2F%282%28x%2By%29%29+-+y++++%29%2A%22y%27%22 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Simplifying, we obtain\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+2xy+-+y%5E2+=+%28x%5E2+-+2xy+-+y%5E2%29%2A%22y%27%22 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+2xy+-+y%5E2+=+%28x%5E2+-+2xy+-+y%5E2%29%2A%22y%27%22\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or finally,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28+%22y%27%22+=+%28x%5E2+%2B+2xy+-+y%5E2+%29%2F%28+x%5E2+-+2xy+-+y%5E2+%29%29 ALIGN=MIDDLE  ALT=\"green%28+%22y%27%22+=+%28x%5E2+%2B+2xy+-+y%5E2+%29%2F%28+x%5E2+-+2xy+-+y%5E2+%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, \r<BR>\n" );
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document.write( "and that is the differential equation representing all circles with center on the line y = x and passing through the origin.  \r<BR>\n" );
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document.write( "I leave the drawing part up to you.  You should be capable enough of doing that, if you were qualified enough to take that course on ODE's.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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