document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1080724>Question 1080724</A>: <I><SPAN STYLE=\"word-break: break-all;\"> please help!!!\r<BR>\n" );
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document.write( "Find the equation for the tangent lines, and the normal lines, to the hyperbolas y^2/4-x^2/2=1 when x=4. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #694829 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><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=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=694829\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The slope of the tangent line is equal to the value of the derivative at that point.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2%2F4-x%5E2%2F2=1 ALIGN=MIDDLE  ALT=\"y%5E2%2F4-x%5E2%2F2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2-2x%5E2=4 ALIGN=MIDDLE  ALT=\"y%5E2-2x%5E2=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Implicitly differentiate,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2ydy-4xdx=0 ALIGN=MIDDLE  ALT=\"2ydy-4xdx=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2ydy=4xdx ALIGN=MIDDLE  ALT=\"2ydy=4xdx\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx=2%28x%2Fy%29 ALIGN=MIDDLE  ALT=\"dy%2Fdx=2%28x%2Fy%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So then,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2-2%284%29%5E2=4 ALIGN=MIDDLE  ALT=\"y%5E2-2%284%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=y%5E2-32=4 ALIGN=MIDDLE  ALT=\"y%5E2-32=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2=36 ALIGN=MIDDLE  ALT=\"y%5E2=36\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=6 ALIGN=MIDDLE  ALT=\"y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-6 ALIGN=MIDDLE  ALT=\"y=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So for (4,6),<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=dy%2Fdx=2%284%2F6%29=4%2F3 ALIGN=MIDDLE  ALT=\"m=dy%2Fdx=2%284%2F6%29=4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using the point slope form of a line,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-6=%284%2F3%29%28x-4%29 ALIGN=MIDDLE  ALT=\"y-6=%284%2F3%29%28x-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-6=%284%2F3%29x-16%2F3%7D%7D%0D%0A%7B%7B%7By=%284%2F3%29x-16%2F3%2B18%2F3 ALIGN=MIDDLE  ALT=\"y-6=%284%2F3%29x-16%2F3%7D%7D%0D%0A%7B%7B%7By=%284%2F3%29x-16%2F3%2B18%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%284%2F3%29x%2B2%2F3 ALIGN=MIDDLE  ALT=\"y=%284%2F3%29x%2B2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and for (4,-6)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=-4%2F3 ALIGN=MIDDLE  ALT=\"m=-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B6=-%284%2F3%29%28x-4%29 ALIGN=MIDDLE  ALT=\"y%2B6=-%284%2F3%29%28x-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B6=-%284%2F3%29x%2B16%2F3%29 ALIGN=MIDDLE  ALT=\"y%2B6=-%284%2F3%29x%2B16%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%284%2F3%29x%2B16%2F3-18%2F3%29 ALIGN=MIDDLE  ALT=\"y=-%284%2F3%29x%2B16%2F3-18%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%284%2F3%29x-2%2F3 ALIGN=MIDDLE  ALT=\"y=-%284%2F3%29x-2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To get the normal lines, you know that the tangent and normal lines are perpendicular to each other.<BR>\n" );
document.write( "So the slopes are negative reciprocals,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5B1%5D%2A%284%2F3%29=-1 ALIGN=MIDDLE  ALT=\"m%5B1%5D%2A%284%2F3%29=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5B1%5D=-3%2F4 ALIGN=MIDDLE  ALT=\"m%5B1%5D=-3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5B2%5D%2A%28-4%2F3%29=-1 ALIGN=MIDDLE  ALT=\"m%5B2%5D%2A%28-4%2F3%29=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5B2%5D=3%2F4 ALIGN=MIDDLE  ALT=\"m%5B2%5D=3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So then,<BR>\n" );
document.write( "(4,6)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-6=%283%2F4%29%28x-4%29 ALIGN=MIDDLE  ALT=\"y-6=%283%2F4%29%28x-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(4,-6)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B6=%283%2F4%29%28x-4%29 ALIGN=MIDDLE  ALT=\"y%2B6=%283%2F4%29%28x-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I leave those to you to put into slope-intercept form.\r<BR>\n" );
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document.write( ".<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/display-illustration.mpl?tutor=Fombitz&illustration_name=234.JPG\">.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );