document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=985482>Question 985482</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For the circle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5E2+%2B+y%5E2+=+4+ ALIGN=MIDDLE  ALT=\"+x%5E2+%2B+y%5E2+=+4+\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y+=+mx+%2B+4+ ALIGN=MIDDLE  ALT=\"+y+=+mx+%2B+4+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, determine the exact values of the gradient <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+m+ ALIGN=MIDDLE  ALT=\"+m+\"   DISABLED_style= \"max-width:90%; height:auto;\" > so that the line:\r<BR>\n" );
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document.write( "a) is a tangent to the circle<BR>\n" );
document.write( "b) intersects the circle in two places<BR>\n" );
document.write( "c) does not intersect the circle </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #606353 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=606353\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+system%28x%5E2+%2B+y%5E2+=+4%2C+y+=+mx+%2B+4%29+ ALIGN=MIDDLE  ALT=\"+system%28x%5E2+%2B+y%5E2+=+4%2C+y+=+mx+%2B+4%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives us the intersection points of circle and line.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+system%28x%5E2+%2B+y%5E2+=+4%2C+y+=+mx+%2B+4%29+ ALIGN=MIDDLE  ALT=\"+system%28x%5E2+%2B+y%5E2+=+4%2C+y+=+mx+%2B+4%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+%28mx+%2B+4%29%5E2+=+4 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+%28mx+%2B+4%29%5E2+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+m%5E2x%5E2+%2B8mx%2B16+=+4 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+m%5E2x%5E2+%2B8mx%2B16+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28m%5E2%2B1%29x%5E2+%2B8mx%2B12+=0 ALIGN=MIDDLE  ALT=\"%28m%5E2%2B1%29x%5E2+%2B8mx%2B12+=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That is a quadratic equation.<BR>\n" );
document.write( "A quadratic equation of the form <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;\" ><BR>\n" );
document.write( "has no real solutions if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4%2Aa%2Ac%3C0 ALIGN=MIDDLE  ALT=\"b%5E2-4%2Aa%2Ac%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "It has one real solution if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4%2Aa%2Ac=0 ALIGN=MIDDLE  ALT=\"b%5E2-4%2Aa%2Ac=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "it has two real solutions if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4%2Aa%2Ac%3E0 ALIGN=MIDDLE  ALT=\"b%5E2-4%2Aa%2Ac%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For the quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28m%5E2%2B1%29x%5E2+%2B8mx%2B12+=0 ALIGN=MIDDLE  ALT=\"%28m%5E2%2B1%29x%5E2+%2B8mx%2B12+=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a=m%5E2%2B1%2Cb=8m%2Cc=12%29 ALIGN=MIDDLE  ALT=\"system%28a=m%5E2%2B1%2Cb=8m%2Cc=12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4%2Aa%2Ac=%288m%29%5E2-4%28m%5E2%2B1%29%2A12=64m%5E2-48m%5E2-48=16m%5E2-48=16%28m%5E2-3%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The zeros of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16%28m%5E2-3%29 ALIGN=MIDDLE  ALT=\"16%28m%5E2-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > are the zeros of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2-3 ALIGN=MIDDLE  ALT=\"m%5E2-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , given by<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=-sqrt%283%29 ALIGN=MIDDLE  ALT=\"m=-sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=sqrt%283%29 ALIGN=MIDDLE  ALT=\"m=sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For those values, the line is tangent to the circle.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-5%2C5%2C-4%2C6%2C%0D%0Agrid%280%29%2Ccircle%280%2C0%2C2%29%2C%0D%0Aline%28-5%2C-4.66%2C5%2C12.66%29%2C%0D%0Aline%28-5%2C12.66%2C5%2C-4.66%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In between those values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sqrt%283%29%3Cm%3Csqrt%283%29 ALIGN=MIDDLE  ALT=\"-sqrt%283%29%3Cm%3Csqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16%28m%5E2-3%29 ALIGN=MIDDLE  ALT=\"16%28m%5E2-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is negative, and the line does not intersect the circle.<BR>\n" );
document.write( "Otherwise, for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28m%3C-sqrt%283%29%2C%22or%22%2Cm%3Esqrt%283%29%29 ALIGN=MIDDLE  ALT=\"system%28m%3C-sqrt%283%29%2C%22or%22%2Cm%3Esqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16%28m%5E2-3%29%3E0%29 ALIGN=MIDDLE  ALT=\"16%28m%5E2-3%29%3E0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and the line intersects the circle  in two places. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );