document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=966011>Question 966011</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A circle has equation x^2+y^2-2x+4y-15=0 <BR>\n" );
document.write( "Find the values for which the line mx+2y-7=0 is a tangent to this circle  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #590476 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=590476\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A circle has equation x^2+y^2-2x+4y-15=0 <BR>\n" );
document.write( "Find the values for which the line mx+2y-7=0 is a tangent to this circle <BR>\n" );
document.write( "-------------<BR>\n" );
document.write( "x^2+y^2-2x+4y-15=0<BR>\n" );
document.write( "(x-1)^2 + (y+2)^2 = 20 is the circle.<BR>\n" );
document.write( "mx+2y-7=0<BR>\n" );
document.write( "y = mx/2 + 3.5<BR>\n" );
document.write( "A point on the tangent lines (2 of them) is A(0,3.5)<BR>\n" );
document.write( "-------------<BR>\n" );
document.write( "The center of the circle is C(1,-2)<BR>\n" );
document.write( "The distance AC = <BR>\n" );
document.write( "The tangent point is P<BR>\n" );
document.write( "CP = r = sqrt(20)<BR>\n" );
document.write( "ACP is a right triangle<BR>\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( "---------------<BR>\n" );
document.write( "Draw a circle centered at A with r^2 = 11.25<BR>\n" );
document.write( "--> x^2 + (y-3.5)^2 = 11.25<BR>\n" );
document.write( "The 2 points of intersection of the 2 circles are the tangent points.<BR>\n" );
document.write( "x^2 + (y-3.5)^2 = 11.25<BR>\n" );
document.write( "x^2 + y^2 - 7y + 12.25 = 11.25<BR>\n" );
document.write( "x^2 + y^2 - 7y = -1<BR>\n" );
document.write( "x^2 + y^2 - 2x + 4y = 15<BR>\n" );
document.write( "------------------------------------------ Subtract<BR>\n" );
document.write( "2x - 11y = -16 (the eqn of the line thru the 2 tangent points)<BR>\n" );
document.write( "Find the intersection of the line and either circle. <BR>\n" );
document.write( "2x - 11y = -16<BR>\n" );
document.write( "y = (2x + 16)/11<BR>\n" );
document.write( "(x-1)^2 + (y+2)^2 = 20<BR>\n" );
document.write( "Sub for y<BR>\n" );
document.write( "(x-1)^2 + ((2x+38)/11)^2 = 20<BR>\n" );
document.write( "121x^2 - 242x + 121 + 4x^2 + 152x + 1444 = 2420<BR>\n" );
document.write( "125x^2 - 90x - 855 = 0<BR>\n" );
document.write( "<TABLE  BORDER=1>\n" );
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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=125x%5E2%2B-90x%2B-855+=+0 ALIGN=MIDDLE  ALT=\"125x%5E2%2B-90x%2B-855+=+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=%28-90%29%5E2-4%2A125%2A-855=435600 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-90%29%5E2-4%2A125%2A-855=435600\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=435600 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--90%2B-sqrt%28+435600+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--90%2B-sqrt%28+435600+%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-%28-90%29%2Bsqrt%28+435600+%29%29%2F2%5C125+=+3 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-90%29%2Bsqrt%28+435600+%29%29%2F2%5C125+=+3\"   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-%28-90%29-sqrt%28+435600+%29%29%2F2%5C125+=+-2.28 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-90%29-sqrt%28+435600+%29%29%2F2%5C125+=+-2.28\"   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=125x%5E2%2B-90x%2B-855 ALIGN=MIDDLE  ALT=\"125x%5E2%2B-90x%2B-855\"   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=125x%5E2%2B-90x%2B-855+=+%28x-3%29%2A%28x--2.28%29 ALIGN=MIDDLE  ALT=\"125x%5E2%2B-90x%2B-855+=+%28x-3%29%2A%28x--2.28%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 3, -2.28.\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+125%2Ax%5E2%2B-90%2Ax%2B-855+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+125%2Ax%5E2%2B-90%2Ax%2B-855+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
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document.write( "==============<BR>\n" );
document.write( "x = -2.28 --> y = 1.04 --> (-2.28,1.04)<BR>\n" );
document.write( "x = 3 --> y = 2 --> (3,2)<BR>\n" );
document.write( "Can you do the rest? </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );