document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=783735>Question 783735</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find  an equation of a circle tangent to 3x+4y-16=0 at (4,1) with a radius of 5. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #476900 by </B><A HREF=/tutors/aboutme.mpl?userid=Alan3354><B>Alan3354(69443)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Alan3354><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=Alan3354><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=476900\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Find  an equation of a circle tangent to 3x+4y-16=0 at (4,1) with a radius of 5.<BR>\n" );
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document.write( "The line thru the center of the circle is perpendicular to the given line, and the center is 5 units from the tangent points.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "Find the slope of the line<BR>\n" );
document.write( "3x+ 4y = 16<BR>\n" );
document.write( "solve for y<BR>\n" );
document.write( "y = -3x/4 + 4<BR>\n" );
document.write( "Slope of the line = -3/4<BR>\n" );
document.write( "Slope of the perpendicular line = 4/3<BR>\n" );
document.write( "---<BR>\n" );
document.write( "Find the eqn of the perpendicular line thru (4,1)<BR>\n" );
document.write( "use y = mx + b and the point to find b, the y-intercept<BR>\n" );
document.write( "1 = (4/3)*4 + b<BR>\n" );
document.write( "b = -13/3<BR>\n" );
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document.write( "The perpendicular line is y = 4x/3 - 13/3<BR>\n" );
document.write( "---<BR>\n" );
document.write( "There are 2 points 5 units from (4,1) on y = 4x/3 - 13/3 --> 2 tangent circles<BR>\n" );
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document.write( "Draw a circle radius = 5 centered at (4,1)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-4%29%5E2+%2B+%28y-1%29%5E2+=+25 ALIGN=MIDDLE  ALT=\"%28x-4%29%5E2+%2B+%28y-1%29%5E2+=+25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "y = 4x/3 - 13/3<BR>\n" );
document.write( "Sub for y<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-4%29%5E2+%2B+%284x%2F3+-+13%2F3+-1%29%5E2+=+25 ALIGN=MIDDLE  ALT=\"%28x-4%29%5E2+%2B+%284x%2F3+-+13%2F3+-1%29%5E2+=+25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Solve for x<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%28x-4%29%5E2+%2B+%284x+-+16%29%5E2+=+225 ALIGN=MIDDLE  ALT=\"9%28x-4%29%5E2+%2B+%284x+-+16%29%5E2+=+225\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E2+-+72x+%2B+144+%2B+16x%5E2+-+128x+%2B+256+=+225 ALIGN=MIDDLE  ALT=\"9x%5E2+-+72x+%2B+144+%2B+16x%5E2+-+128x+%2B+256+=+225\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25x%5E2+-+200x+%2B+175+=+0 ALIGN=MIDDLE  ALT=\"25x%5E2+-+200x+%2B+175+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=solve_quadratic_equation.solver>SOLVE quadratic equation (work shown, graph etc)</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25x%5E2%2B-200x%2B175+=+0 ALIGN=MIDDLE  ALT=\"25x%5E2%2B-200x%2B175+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%28-200%29%5E2-4%2A25%2A175=22500 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-200%29%5E2-4%2A25%2A175=22500\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=22500 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28--200%2B-sqrt%28+22500+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--200%2B-sqrt%28+22500+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+%28-%28-200%29%2Bsqrt%28+22500+%29%29%2F2%5C25+=+7 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-200%29%2Bsqrt%28+22500+%29%29%2F2%5C25+=+7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%28-200%29-sqrt%28+22500+%29%29%2F2%5C25+=+1 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-200%29-sqrt%28+22500+%29%29%2F2%5C25+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25x%5E2%2B-200x%2B175 ALIGN=MIDDLE  ALT=\"25x%5E2%2B-200x%2B175\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25x%5E2%2B-200x%2B175+=+%28x-7%29%2A%28x-1%29 ALIGN=MIDDLE  ALT=\"25x%5E2%2B-200x%2B175+=+%28x-7%29%2A%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 7, 1.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+25%2Ax%5E2%2B-200%2Ax%2B175+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+25%2Ax%5E2%2B-200%2Ax%2B175+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "\n" );
document.write( "==============<BR>\n" );
document.write( "x = 7, y = 5 --> (7,5)<BR>\n" );
document.write( "--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-7%29%5E2+%2B+%28y-5%29%5E2+=+25 ALIGN=MIDDLE  ALT=\"%28x-7%29%5E2+%2B+%28y-5%29%5E2+=+25\"   DISABLED_style= \"max-width:90%; height:auto;\" > is one circle<BR>\n" );
document.write( "========================================<BR>\n" );
document.write( "x = 1, y = -3 --> (1,-3)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%5E2+%2B+%28y%2B3%29%5E2+=+25 ALIGN=MIDDLE  ALT=\"%28x-1%29%5E2+%2B+%28y%2B3%29%5E2+=+25\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the 2nd circle </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );