document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=672756>Question 672756</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find an equation(s) of the line(s) containing (5,4) and at a distance 2 from (-1,-3). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #418308 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=418308\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Find an equation(s) of the line(s) containing (5,4) and at a distance 2 from (-1,-3).<BR>\n" );
document.write( "-----------<BR>\n" );
document.write( "The lines are tangents to a circle of radius 2 centered at (-1,-3)<BR>\n" );
document.write( "The distance from (5,4) to the center (-1,-3) = sqrt(85).<BR>\n" );
document.write( "Right angles are formed at the tangent points.<BR>\n" );
document.write( "----<BR>\n" );
document.write( "The distance from (5,4) to the tangent points = 9.<BR>\n" );
document.write( "The tangent points are the intersection of the circle above and a circle of radius 9 centered at (5,4).<BR>\n" );
document.write( "----------<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B1%29%5E2+%2B+%28y%2B3%29%5E2+=+4 ALIGN=MIDDLE  ALT=\"%28x%2B1%29%5E2+%2B+%28y%2B3%29%5E2+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-5%29%5E2+%2B+%28y-4%29%5E2+=+81 ALIGN=MIDDLE  ALT=\"%28x-5%29%5E2+%2B+%28y-4%29%5E2+=+81\"   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%5E2+%2B+y%5E2+%2B+2x+%2B+6y+=+-6 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+y%5E2+%2B+2x+%2B+6y+=+-6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+y%5E2+-+10x+-+8y+=+40 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+y%5E2+-+10x+-+8y+=+40\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "----------------------------------- Subtract<BR>\n" );
document.write( "12x + 14y = -46 <BR>\n" );
document.write( "6x + 7y = -23 is an equation of the line thru the 2 tangent points.<BR>\n" );
document.write( "y = (-6x - 23)/7<BR>\n" );
document.write( "Sub for y in one of the circles<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+y%5E2+%2B+2x+%2B+6y+=+-6 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+y%5E2+%2B+2x+%2B+6y+=+-6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+%2836x%5E2+%2B+276x+%2B+529%29%2F49+%2B+2x+%2B+%28-36x+-+138%29%2F7+=+-6 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+%2836x%5E2+%2B+276x+%2B+529%29%2F49+%2B+2x+%2B+%28-36x+-+138%29%2F7+=+-6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=49x%5E2+%2B+36x%5E2+%2B+276x+%2B+529+%2B+98x+-+252x+-+966+=+-294 ALIGN=MIDDLE  ALT=\"49x%5E2+%2B+36x%5E2+%2B+276x+%2B+529+%2B+98x+-+252x+-+966+=+-294\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=85x%5E2+%2B+122x+-+143+=+0 ALIGN=MIDDLE  ALT=\"85x%5E2+%2B+122x+-+143+=+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=85x%5E2%2B122x%2B-143+=+0 ALIGN=MIDDLE  ALT=\"85x%5E2%2B122x%2B-143+=+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=%28122%29%5E2-4%2A85%2A-143=63504 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28122%29%5E2-4%2A85%2A-143=63504\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=63504 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+=+%28-122%2B-sqrt%28+63504+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-122%2B-sqrt%28+63504+%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-%28122%29%2Bsqrt%28+63504+%29%29%2F2%5C85+=+0.764705882352941 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28122%29%2Bsqrt%28+63504+%29%29%2F2%5C85+=+0.764705882352941\"   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-%28122%29-sqrt%28+63504+%29%29%2F2%5C85+=+-2.2 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28122%29-sqrt%28+63504+%29%29%2F2%5C85+=+-2.2\"   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=85x%5E2%2B122x%2B-143 ALIGN=MIDDLE  ALT=\"85x%5E2%2B122x%2B-143\"   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=85x%5E2%2B122x%2B-143+=+%28x-0.764705882352941%29%2A%28x--2.2%29 ALIGN=MIDDLE  ALT=\"85x%5E2%2B122x%2B-143+=+%28x-0.764705882352941%29%2A%28x--2.2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 0.764705882352941, -2.2.\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+85%2Ax%5E2%2B122%2Ax%2B-143+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+85%2Ax%5E2%2B122%2Ax%2B-143+%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 = -2.2 --> y = (-6*-2.2 - 23)/7 = -1.4<BR>\n" );
document.write( "Tangent point at (-2.2,-1.4)<BR>\n" );
document.write( "Equation of the line thru (-2.2,-1.4) and (5,4) is <BR>\n" );
document.write( "3x - 4y = -1<BR>\n" );
document.write( "==============================================<BR>\n" );
document.write( "x = 0.7647 --> y = -3.94117 --> tangent point at (0.7647,-3.84117)<BR>\n" );
document.write( "Equation of line thru the 2 points is<BR>\n" );
document.write( "7.94x - 4.2353y = 22.7588 (approximation) </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );