document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191708>Question 1191708</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The equation of a curve is y = x^2/(x+2). The tangent to the curve at the point where x = -3 meets the<BR>\n" );
document.write( "y- axis at M. The normal to the curve at the point where x = -3 meets the x - axis at N. Find the<BR>\n" );
document.write( "area of the triangle MNO, where O is the origin. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #823562 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=823562\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We first locate the precise point on the curve where the tangent and normal lines are:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28+-+3%29=x%5E2+%2F+%28x+%2B+2%29+=+%28-3%29%5E2+%2F+%28-3+%2B+2%29+=+9%2F-1+=+-9 ALIGN=MIDDLE  ALT=\"f%28+-+3%29=x%5E2+%2F+%28x+%2B+2%29+=+%28-3%29%5E2+%2F+%28-3+%2B+2%29+=+9%2F-1+=+-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the point we’re looking for is (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9 ALIGN=MIDDLE  ALT=\"-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >). Next, we find the derivative of the curve, and determine its slope at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-3 ALIGN=MIDDLE  ALT=\"x+=+-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >’=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%2B4%29%2F%28x%2B2%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%5E2%2B4%29%2F%28x%2B2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ; -><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f ALIGN=MIDDLE  ALT=\"f\"   DISABLED_style= \"max-width:90%; height:auto;\" >’<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-+3%29=+-+3 ALIGN=MIDDLE  ALT=\"%28-+3%29=+-+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Equation of tangent at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=+-3 ALIGN=MIDDLE  ALT=\"x=+-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is : <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=+-+3x+-18 ALIGN=MIDDLE  ALT=\"y=+-+3x+-18\"   DISABLED_style= \"max-width:90%; height:auto;\" > ; M lie on y-axis, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >then the other coordinate of M is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=+-+3%280%29+-18=-18 ALIGN=MIDDLE  ALT=\"y=+-+3%280%29+-18=-18\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so M(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ; - 18)\r<BR>\n" );
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document.write( "So the slope of the tangent line is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, while the slope of the normal is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F-3=1%2F3 ALIGN=MIDDLE  ALT=\"-1%2F-3=1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Equation of normal at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=+-3 ALIGN=MIDDLE  ALT=\"x=+-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is ; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=++%281%2F3%29x+-+8+ ALIGN=MIDDLE  ALT=\"y+=++%281%2F3%29x+-+8+\"   DISABLED_style= \"max-width:90%; height:auto;\" >; the x- coordinate of N is intersection of a line with x-axis\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=++%281%2F3%29x+-+8+ ALIGN=MIDDLE  ALT=\"0=++%281%2F3%29x+-+8+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=24+ ALIGN=MIDDLE  ALT=\"x=24+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "N(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24 ALIGN=MIDDLE  ALT=\"24\"   DISABLED_style= \"max-width:90%; height:auto;\" > ; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "In right triangle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=MON+ ALIGN=MIDDLE  ALT=\"MON+\"   DISABLED_style= \"max-width:90%; height:auto;\" >: ; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=MO=18 ALIGN=MIDDLE  ALT=\"MO=18\"   DISABLED_style= \"max-width:90%; height:auto;\" > ; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=NO=24 ALIGN=MIDDLE  ALT=\"NO=24\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "then the area <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S ALIGN=MIDDLE  ALT=\"S\"   DISABLED_style= \"max-width:90%; height:auto;\" > is :\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=%281%2F2%29%2AMO%2ANO ALIGN=MIDDLE  ALT=\"S=%281%2F2%29%2AMO%2ANO\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=%281%2F2%29%2A18%2A24 ALIGN=MIDDLE  ALT=\"S=%281%2F2%29%2A18%2A24\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=216 ALIGN=MIDDLE  ALT=\"S=216\"   DISABLED_style= \"max-width:90%; height:auto;\" > square units\r<BR>\n" );
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