document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=677579>Question 677579</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A question: Show that the line x+16y+209=0 is normal to the curve y=10x-3x^2 at one of the points of intersection, and find this point? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #421414 by </B><A HREF=/tutors/aboutme.mpl?userid=lwsshak3><B>lwsshak3(11628)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=lwsshak3><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=421414\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Show that the line x+16y+209=0 is normal to the curve y=10x-3x^2 at one of the points of intersection, and find this point?<BR>\n" );
document.write( "**<BR>\n" );
document.write( "y=10x-3x^2<BR>\n" );
document.write( "x+16y+209=0<BR>\n" );
document.write( "..<BR>\n" );
document.write( "x+16(10x-3x^2)+209=0<BR>\n" );
document.write( "x+160x-48x^2+209=0<BR>\n" );
document.write( "48x^2-161x-209=0<BR>\n" );
document.write( "solve for x by quadratic formula as follows:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "a=48, b=-161, c=-209<BR>\n" );
document.write( "ans: <BR>\n" );
document.write( "x=-1 (At one of the points of intersection)<BR>\n" );
document.write( "y=10x-3x2=-10-3=-13<BR>\n" );
document.write( "point of intersection: (-1,-13)<BR>\n" );
document.write( "or<BR>\n" );
document.write( "x&#8776;4.35417<BR>\n" );
document.write( "..<BR>\n" );
document.write( "Take derivative of first equation:y=10x-3x^2<BR>\n" );
document.write( "y'=10-6x (this is an equation of slope as a function of x)<BR>\n" );
document.write( "slope at point of intersection=10-6(-1)=16<BR>\n" );
document.write( "..<BR>\n" );
document.write( "x+16y+209=0<BR>\n" );
document.write( "16y=-x-209<BR>\n" );
document.write( "y=-x/16-209/16<BR>\n" );
document.write( "slope=-1/16<BR>\n" );
document.write( "This slope is the negative reciprocal of the slope of the curve at the point of intersection<BR>\n" );
document.write( "Therefore, equation of the line,x+16y+209=0, is normal to the curve, y=10x-3x^2,at the point of intersection(-1,-13) </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );