document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=823766>Question 823766</A>: <I><SPAN STYLE=\"word-break: break-all;\"> the equations of the tangents to ellipse 9x^2+16y^2=144 from point (2,3) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #495797 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=495797\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'm assuming that you are in a Calculus class. If I am wrong then you'll have to re-post (and ask for a non-Calculus solution).<br><BR>\n" );
document.write( "9x^2+16y^2=144<BR>\n" );
document.write( "To find the equation of the tangent(s), we will need the slope(s). For the slopes we will use the first derivative. And since it is a lot of work to rewrite this equation as two functions of x (one for the upper half of the ellipse and one for the lower half), we will use implicit differentiation:<BR>\n" );
document.write( "2*9*x^(2-1) + 2*16*y^(2-1)*y' = 0<BR>\n" );
document.write( "which simplifies to:<BR>\n" );
document.write( "18x + 32y*y' = 0<BR>\n" );
document.write( "Solving for y':<BR>\n" );
document.write( "32y*y' = -18x<BR>\n" );
document.write( "y' = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-18x%29%2F32y ALIGN=MIDDLE  ALT=\"%28-18x%29%2F32y\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which reduces to:<BR>\n" );
document.write( "y' = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-9x%29%2F16y ALIGN=MIDDLE  ALT=\"%28-9x%29%2F16y\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now we can use this to find the slope(s) of the tangent(s) at (2, 3):<BR>\n" );
document.write( "y' = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-9%282%29%29%2F16%283%29 ALIGN=MIDDLE  ALT=\"%28-9%282%29%29%2F16%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies:<BR>\n" );
document.write( "y' = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-18%29%2F48 ALIGN=MIDDLE  ALT=\"%28-18%29%2F48\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which reduces:<BR>\n" );
document.write( "y' = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-3%29%2F8 ALIGN=MIDDLE  ALT=\"%28-3%29%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since we found only one slope, there is only one tangent. It has a slope of -3/8 and passes through (2, 3). There are a few ways to use this data to find the equation. I like the point-slope form:<BR>\n" );
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;\" ><BR>\n" );
document.write( "Substituting our slope and the coordinates of the point we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+-+%283%29+=+%28%28-3%29%2F8%29%28x-%282%29%29 ALIGN=MIDDLE  ALT=\"y+-+%283%29+=+%28%28-3%29%2F8%29%28x-%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+-+3+=+%28%28-3%29%2F8%29%28x-2%29 ALIGN=MIDDLE  ALT=\"y+-+3+=+%28%28-3%29%2F8%29%28x-2%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+-+3+=+%28%28-3%29%2F8%29x-%28-3%29%2F4 ALIGN=MIDDLE  ALT=\"y+-+3+=+%28%28-3%29%2F8%29x-%28-3%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+-+3+=+%28%28-3%29%2F8%29x%2B3%2F4 ALIGN=MIDDLE  ALT=\"y+-+3+=+%28%28-3%29%2F8%29x%2B3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This may be an acceptable solution. Or you could change it into either standard form, Ax + By = C, or slope-intercept form, y = mx + b. I'll leave it up to you to decide<ul><li>if you need the equation in a certain form and if so...</li><li>which form you want and...</li><li>how to transform the equation we have into that form.</li></ul>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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