document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1083556>Question 1083556</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For what value of k , line 3x+4y+k=0 is tangent to the circle x^2+y^2+6x-y-1=0?<BR>\n" );
document.write( "a)13<BR>\n" );
document.write( "a)19<BR>\n" );
document.write( "c)25<BR>\n" );
document.write( "d)30 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #697581 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=697581\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> For what value of k , line 3x+4y+k=0 is tangent to the circle x^2+y^2+6x-y-1=0?<BR>\n" );
document.write( "------------<BR>\n" );
document.write( "3x+4y+k=0<BR>\n" );
document.write( "y = (-3/4)x - k/4<BR>\n" );
document.write( "Slope m = -3/4<BR>\n" );
document.write( "================<BR>\n" );
document.write( "x^2+y^2+6x-y-1=0<BR>\n" );
document.write( "x^2 + 6x + 9 + y^2 - y + 1/4 = 1 + 9 + 1/4 = 10.25<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B3%29%5E2+%2B+%28y-1%2F2%29%5E2+=+10.25 ALIGN=MIDDLE  ALT=\"%28x%2B3%29%5E2+%2B+%28y-1%2F2%29%5E2+=+10.25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The center is (-3,1/2)<BR>\n" );
document.write( "Find the line thru the center perpendicular to the given line.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "slope = -1/m = 4/3<BR>\n" );
document.write( "y- 1/2 = (4/3)*(x+3)<BR>\n" );
document.write( "The 2 tangent points are the intersection of y- 1/2 = (4/3)*(x+3) and the circle.<BR>\n" );
document.write( "----<BR>\n" );
document.write( "2y- 1 = (8/3)*(x+3) = 8x/3 + 8<BR>\n" );
document.write( "y = 4x/3 + 9/2<BR>\n" );
document.write( "---<BR>\n" );
document.write( "x^2+y^2+6x-y-1=0<BR>\n" );
document.write( "Sub for y<BR>\n" );
document.write( "x^2 + (4x/3 + 9/2)^2 + 6x - (4x/3 + 9/2) - 1 = 0<BR>\n" );
document.write( "x^2 + 16x^2/9 + 12x + 81/4 + 6x - 4x/3 - 9/2 - 1 = 0<BR>\n" );
document.write( "25x^2/9 + 50x/3 + 59/4 = 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=2.77777777777778x%5E2%2B16.6666666666667x%2B14.75+=+0 ALIGN=MIDDLE  ALT=\"2.77777777777778x%5E2%2B16.6666666666667x%2B14.75+=+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=%2816.6666666666667%29%5E2-4%2A2.77777777777778%2A14.75=113.88888888889 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%2816.6666666666667%29%5E2-4%2A2.77777777777778%2A14.75=113.88888888889\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=113.88888888889 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+=+%28-16.6666666666667%2B-sqrt%28+113.88888888889+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-16.6666666666667%2B-sqrt%28+113.88888888889+%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-%2816.6666666666667%29%2Bsqrt%28+113.88888888889+%29%29%2F2%5C2.77777777777778+=+-1.07906272877014 ALIGN=MIDDLE   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-%2816.6666666666667%29-sqrt%28+113.88888888889+%29%29%2F2%5C2.77777777777778+=+-4.92093727122987 ALIGN=MIDDLE   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=2.77777777777778x%5E2%2B16.6666666666667x%2B14.75 ALIGN=MIDDLE  ALT=\"2.77777777777778x%5E2%2B16.6666666666667x%2B14.75\"   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=2.77777777777778x%5E2%2B16.6666666666667x%2B14.75+=+%28x--1.07906272877014%29%2A%28x--4.92093727122987%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: -1.07906272877014, -4.92093727122987.\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+2.77777777777778%2Ax%5E2%2B16.6666666666667%2Ax%2B14.75+%29 ALIGN=MIDDLE   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( "find the y values, then the equations of the lines thru the 2 points with a slope of -3/4.<BR>\n" );
document.write( "then find 2 values for k.\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );