document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=940984>Question 940984</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the equation of a circle centre on the line y=2x+1 touching the y axis and passing through A(4,5) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #573562 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=573562\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Find the equation of a circle centre on the line y=2x+1 touching the y axis and passing through A(4,5)<BR>\n" );
document.write( "===========<BR>\n" );
document.write( "If by touching you mean tangent:<BR>\n" );
document.write( "--<BR>\n" );
document.write( "Its center is equidistant from (4,5) and the y-axis.<BR>\n" );
document.write( "The center is (h,k)<BR>\n" );
document.write( "The distance to (4,5) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28h-4%29%5E2+%2B+%28k-5%29%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28h-4%29%5E2+%2B+%28k-5%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The distance to the y-axis = h<BR>\n" );
document.write( "y = 2x + 1 --> k = 2h + 1<BR>\n" );
document.write( "---<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+=+%28h-4%29%5E2+%2B+%28k-5%29%5E2 ALIGN=MIDDLE  ALT=\"h%5E2+=+%28h-4%29%5E2+%2B+%28k-5%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%5E2+=+h%5E2-8h%2B16+%2B+k%5E2-10k%2B25 ALIGN=MIDDLE  ALT=\"h%5E2+=+h%5E2-8h%2B16+%2B+k%5E2-10k%2B25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8h%2B16+%2B+k%5E2-10k%2B25+=+0 ALIGN=MIDDLE  ALT=\"-8h%2B16+%2B+k%5E2-10k%2B25+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Sub for k<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8h%2B16+%2B+%282h%2B1%29%5E2-10%282h%2B1%29%2B25+=+0 ALIGN=MIDDLE  ALT=\"-8h%2B16+%2B+%282h%2B1%29%5E2-10%282h%2B1%29%2B25+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8h%2B16+%2B+4h%5E2%2B4h%2B1-20h-10%2B25+=+0 ALIGN=MIDDLE  ALT=\"-8h%2B16+%2B+4h%5E2%2B4h%2B1-20h-10%2B25+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4h%5E2+-+4h+%2B+17+-+20h+%2B+15+=+0 ALIGN=MIDDLE  ALT=\"4h%5E2+-+4h+%2B+17+-+20h+%2B+15+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4h%5E2+-+24h+%2B+32+=+0 ALIGN=MIDDLE  ALT=\"4h%5E2+-+24h+%2B+32+=+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=4x%5E2%2B-24x%2B32+=+0 ALIGN=MIDDLE  ALT=\"4x%5E2%2B-24x%2B32+=+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=%28-24%29%5E2-4%2A4%2A32=64 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-24%29%5E2-4%2A4%2A32=64\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=64 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--24%2B-sqrt%28+64+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--24%2B-sqrt%28+64+%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-%28-24%29%2Bsqrt%28+64+%29%29%2F2%5C4+=+4 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-24%29%2Bsqrt%28+64+%29%29%2F2%5C4+=+4\"   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-%28-24%29-sqrt%28+64+%29%29%2F2%5C4+=+2 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-24%29-sqrt%28+64+%29%29%2F2%5C4+=+2\"   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=4x%5E2%2B-24x%2B32 ALIGN=MIDDLE  ALT=\"4x%5E2%2B-24x%2B32\"   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=4x%5E2%2B-24x%2B32+=+%28x-4%29%2A%28x-2%29 ALIGN=MIDDLE  ALT=\"4x%5E2%2B-24x%2B32+=+%28x-4%29%2A%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 4, 2.\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+4%2Ax%5E2%2B-24%2Ax%2B32+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+4%2Ax%5E2%2B-24%2Ax%2B32+%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( "h = x = 4  --> center @ (4,9)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-4%29%5E2+%2B+%28y-9%29%5E2+=+16 ALIGN=MIDDLE  ALT=\"%28x-4%29%5E2+%2B+%28y-9%29%5E2+=+16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "=============<BR>\n" );
document.write( "x = 2 --> center @ (2,5)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%5E2+%2B+%28y-5%29%5E2+=+4 ALIGN=MIDDLE  ALT=\"%28x-2%29%5E2+%2B+%28y-5%29%5E2+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );