document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1066857>Question 1066857</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the equation of the circle which is tangent to the line 3y-4x-11 and pasese through (8,4) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #682117 by </B><A HREF=/tutors/aboutme.mpl?userid=math_helper><B>math_helper(2461)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_helper><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=math_helper><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=682117\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The problem, as stated, has an infinite number of solutions.  \r<BR>\n" );
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document.write( "I will find one circle then explain why there are an infinite number of solutions.  <BR>\n" );
document.write( "The circle I will find is the one tangent to the line 3y-4x-11=0  AND also tangent to a line parallel to this line where this 2nd line passes through (8,4).  The circle is tangent to this 2nd line at (8,4).   In this way, there is only one solution. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3y+-+4x+-+11+=+0+ ALIGN=MIDDLE  ALT=\"3y+-+4x+-+11+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  —>   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y+=+%284%2F3%29x+%2B+%2811%2F3%29+ ALIGN=MIDDLE  ALT=\"+y+=+%284%2F3%29x+%2B+%2811%2F3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >   \r<BR>\n" );
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document.write( "Now imagine a line passing through (8,4), then the center of the circle, then through <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y=%284%2F3%29x%2B%2811%2F3%29+ ALIGN=MIDDLE  ALT=\"+y=%284%2F3%29x%2B%2811%2F3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > at a point where the line is tangent to the circle.  This (3rd) line has slope -3/4 because it must be perpendicular to  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y+=+%284%2F3%29x+%2B+%2811%2F3%29+ ALIGN=MIDDLE  ALT=\"+y+=+%284%2F3%29x+%2B+%2811%2F3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >   (if a line has slope m then a line perpendicular to it has slope -1/m).\r<BR>\n" );
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document.write( "The line through the center of the circle also passes through (8,4)  so we can find the equation of it.\r<BR>\n" );
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document.write( "                       y = mx+b<BR>\n" );
document.write( "                       4 = (-3/4)(8) + b<BR>\n" );
document.write( "                       4 + (3/4)(8) = b<BR>\n" );
document.write( "                       4 + 6 = b  —> b=10<BR>\n" );
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document.write( "                  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y+=+%28-3%2F4%29x+%2B+10+ ALIGN=MIDDLE  ALT=\"+y+=+%28-3%2F4%29x+%2B+10+\"   DISABLED_style= \"max-width:90%; height:auto;\" >    <<< line through center of circle, <BR>\n" );
document.write( "                                                                 perpendicular to  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y+=+%284%2F3%29x+%2B+%2811%2F3%29+ ALIGN=MIDDLE  ALT=\"+y+=+%284%2F3%29x+%2B+%2811%2F3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >   \r<BR>\n" );
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document.write( "Where these two lines meet, their y values are the same:\r<BR>\n" );
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document.write( "                        <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28-3%2F4%29x+%2B+10+ ALIGN=MIDDLE  ALT=\"+%28-3%2F4%29x+%2B+10+\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%284%2F3%29x+%2B+%2811%2F3%29+ ALIGN=MIDDLE  ALT=\"+%284%2F3%29x+%2B+%2811%2F3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "                         This reduces to  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+=+76%2F25+ ALIGN=MIDDLE  ALT=\"+x+=+76%2F25+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  —>  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y+=+193%2F25+ ALIGN=MIDDLE  ALT=\"+y+=+193%2F25+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the diameter of the circle is:\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+sqrt%28%288+-+76%2F25%29%5E2+%2B+%284+-+193%2F25%29%5E2%29%29+ ALIGN=MIDDLE  ALT=\"+sqrt%28%288+-+76%2F25%29%5E2+%2B+%284+-+193%2F25%29%5E2%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+6.20+ ALIGN=MIDDLE  ALT=\"+6.20+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  ——>   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+r+=+3.10+ ALIGN=MIDDLE  ALT=\"+r+=+3.10+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The center of the circle is at  ( <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%288%2B%2875%2F25%29%29%2F2+ ALIGN=MIDDLE  ALT=\"+%288%2B%2875%2F25%29%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=++%284%2B%28193%2F25%29%29%2F2++ ALIGN=MIDDLE  ALT=\"++%284%2B%28193%2F25%29%29%2F2++\"   DISABLED_style= \"max-width:90%; height:auto;\" > )<BR>\n" );
document.write( "   which works out to  ( <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+5.52+ ALIGN=MIDDLE  ALT=\"+5.52+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+5.86 ALIGN=MIDDLE  ALT=\"+5.86\"   DISABLED_style= \"max-width:90%; height:auto;\" > )<BR>\n" );
document.write( "—\r<BR>\n" );
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document.write( "The equation of the circle is therefore:   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+highlight%28+%28x-5.52%29%5E2+%2B+%28y-5.86%29%5E2+=+%283.10%29%5E2+%29 ALIGN=MIDDLE  ALT=\"+highlight%28+%28x-5.52%29%5E2+%2B+%28y-5.86%29%5E2+=+%283.10%29%5E2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "—\r<BR>\n" );
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document.write( "Now, the original problem statement has an infinite number of solutions because you can make a circle tangent to   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y+=+%284%2F3%29x+%2B+%2811%2F3%29+ ALIGN=MIDDLE  ALT=\"+y+=+%284%2F3%29x+%2B+%2811%2F3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >   that is a little bigger than the one we just found above, and passes through (8,4) but at a different point on the circle than the line perpendicular to  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y+=+%284%2F3%29x+%2B+%2811%2F3%29+ ALIGN=MIDDLE  ALT=\"+y+=+%284%2F3%29x+%2B+%2811%2F3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >.   This can be done an infinite number of times by moving the tangent point (& adjusting circle size) by an infinitesimally small amount.<BR>\n" );
document.write( "—\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+drawing%28300%2C300%2C+-10%2C10%2C-10%2C10%2C%0D%0A++++++++++++++++grid%281%29%2C%0D%0A++++++++++++++++green%28circle%285.52%2C+5.56%2C3.10%29+%29%2C%0D%0A++++++++++++++++line%28-10%2C-9.6667%2C++10%2C+17%29%2C%0D%0A++++++++++++++++line%28-10%2C+17.5%2C+10%2C+2.5%29%2C%0D%0A++++++++++++++++line%280%2C+-20%2F3%2C+9%2C+5.333%29%0D%0A+++++%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "  </SPAN>\n" );
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