document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=243003>Question 243003</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The equation of the tangent line to the curve x^2 + y^2 =169 at the point    (5,-12) is<BR>\n" );
document.write( "(A) 5y-12x= -120<BR>\n" );
document.write( "(B) 5x-12y= 119<BR>\n" );
document.write( "(C) 5x-12y= 169<BR>\n" );
document.write( "(D) 12x+5y= 0<BR>\n" );
document.write( "(E) 12x+5y= 169 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #178018 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=178018\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First derive both sides of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x^2+y^2=169\"> with respect to 'x' to get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x^2+y^2=169\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{d}{dx}\left(x^2+y^2\right)=\frac{d}{dx}\left(169\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2x+2yy^{\prime}=0\">\r<BR>\n" );
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document.write( "Now solve for y':\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2x+2yy^{\prime}=0\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2yy^{\prime}=-2x\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y^{\prime}=\frac{-2x}{2y}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y^{\prime}=-\frac{x}{y}\">\r<BR>\n" );
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document.write( "So the slope is simply the negative quotient of the two coordinates. For the point (5,-12), the slope at that point is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=-5%2F%28-12%29=5%2F12 ALIGN=MIDDLE  ALT=\"m=-5%2F%28-12%29=5%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5 ALIGN=MIDDLE  ALT=\"x=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-12 ALIGN=MIDDLE  ALT=\"y=-12\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "-----------------------------------------------------\r<BR>\n" );
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document.write( "Now recall that the point slope formula is \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-y%5B1%5D=m%28x-x%5B1%5D%29 ALIGN=MIDDLE  ALT=\"y-y%5B1%5D=m%28x-x%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > where 'm' is the slope and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(x_{1},y_{1}\right)\"> is the point in which the line goes through.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y--12=%285%2F12%29%28x-5%29 ALIGN=MIDDLE  ALT=\"y--12=%285%2F12%29%28x-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=5%2F12 ALIGN=MIDDLE  ALT=\"m=5%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D=5 ALIGN=MIDDLE  ALT=\"x%5B1%5D=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D=-12 ALIGN=MIDDLE  ALT=\"y%5B1%5D=-12\"   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%2B12=%285%2F12%29%28x-5%29 ALIGN=MIDDLE  ALT=\"y%2B12=%285%2F12%29%28x-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y--12 ALIGN=MIDDLE  ALT=\"y--12\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B12 ALIGN=MIDDLE  ALT=\"y%2B12\"   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%2B12=%285%2F12%29x%2B%285%2F12%29%28-5%29 ALIGN=MIDDLE  ALT=\"y%2B12=%285%2F12%29x%2B%285%2F12%29%28-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F12 ALIGN=MIDDLE  ALT=\"5%2F12\"   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%2B12=%285%2F12%29x-25%2F12 ALIGN=MIDDLE  ALT=\"y%2B12=%285%2F12%29x-25%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F12 ALIGN=MIDDLE  ALT=\"5%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5 ALIGN=MIDDLE  ALT=\"-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-25%2F12 ALIGN=MIDDLE  ALT=\"-25%2F12\"   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=%285%2F12%29x-25%2F12-12 ALIGN=MIDDLE  ALT=\"y=%285%2F12%29x-25%2F12-12\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 12 from  both sides to isolate y\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%285%2F12%29x-169%2F12 ALIGN=MIDDLE  ALT=\"y=%285%2F12%29x-169%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-25%2F12 ALIGN=MIDDLE  ALT=\"-25%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-12 ALIGN=MIDDLE  ALT=\"-12\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-169%2F12 ALIGN=MIDDLE  ALT=\"-169%2F12\"   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=12y=5x-169 ALIGN=MIDDLE  ALT=\"12y=5x-169\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply EVERY term by the LCD 12 to clear out the fractions.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5x%2B12y=-169 ALIGN=MIDDLE  ALT=\"-5x%2B12y=-169\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 5x from both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x-12y=169 ALIGN=MIDDLE  ALT=\"5x-12y=169\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply EVERY term by -1 to make the 'x' coefficient positive.\r<BR>\n" );
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document.write( "So the equation of the tangent line is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x-12y=169 ALIGN=MIDDLE  ALT=\"5x-12y=169\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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