document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=913268>Question 913268</A>: <I><SPAN STYLE=\"word-break: break-all;\"> On the hyperbola x^2/24-y^2/18=1 find the point M1 nearest to the line 3x+2y+1=0, and compute the distance d from M1 to the line. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #554443 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=554443\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The shortest distance would be a perpendicular line from the line, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B2y%2B1=0 ALIGN=MIDDLE  ALT=\"3x%2B2y%2B1=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2y=-3x-1 ALIGN=MIDDLE  ALT=\"2y=-3x-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=-%283%2F2%29x-1%2F2 ALIGN=MIDDLE  ALT=\"y=-%283%2F2%29x-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( "A point on the hyperbola that had the same slope would be at the minimum distance.<BR>\n" );
document.write( "Find the slope of the hyperbola by differentiating,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2F24-y%5E2%2F18=1 ALIGN=MIDDLE  ALT=\"x%5E2%2F24-y%5E2%2F18=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F24%29%282xdx%29-%281%2F18%29%282ydy%29=0 ALIGN=MIDDLE  ALT=\"%281%2F24%29%282xdx%29-%281%2F18%29%282ydy%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28xdx%29%2F12=%28ydy%29%2F9 ALIGN=MIDDLE  ALT=\"%28xdx%29%2F12=%28ydy%29%2F9\"   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=%283x%29%2F%284y%29=-3%2F2 ALIGN=MIDDLE  ALT=\"dy%2Fdx=%283x%29%2F%284y%29=-3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x=-12y ALIGN=MIDDLE  ALT=\"6x=-12y\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2y ALIGN=MIDDLE  ALT=\"x=-2y\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substitute into the hyperbola equation,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2y%29%5E2%2F24-y%5E2%2F18=1 ALIGN=MIDDLE  ALT=\"%28-2y%29%5E2%2F24-y%5E2%2F18=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%2F6-y%5E2%2F18=1 ALIGN=MIDDLE  ALT=\"y%5E2%2F6-y%5E2%2F18=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2F18%29y%5E2-%281%2F18%29y%5E2=1 ALIGN=MIDDLE  ALT=\"%283%2F18%29y%5E2-%281%2F18%29y%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F18%29y%5E2=1 ALIGN=MIDDLE  ALT=\"%282%2F18%29y%5E2=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=9 ALIGN=MIDDLE  ALT=\"y%5E2=9\"   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 ALIGN=MIDDLE  ALT=\"y=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3 ALIGN=MIDDLE  ALT=\"y=3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then, <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2F24-9%2F18=1 ALIGN=MIDDLE  ALT=\"x%5E2%2F24-9%2F18=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2F24=3%2F2 ALIGN=MIDDLE  ALT=\"x%5E2%2F24=3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=36 ALIGN=MIDDLE  ALT=\"x%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=x=6 ALIGN=MIDDLE  ALT=\"x=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-6 ALIGN=MIDDLE  ALT=\"x=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-10%2C10%2C-10%2C10%2Cgrid%281%29%2Ccircle%28-6%2C3%2C0.3%29%2Ccircle%286%2C-3%2C0.3%29%2Cgraph%28300%2C300%2C-10%2C10%2C-10%2C10%2C-%283%2F2%29x-1%2F2%2C-sqrt%2818%28x%5E2%2F24-1%29%29%2Csqrt%2818%28x%5E2%2F24-1%29%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the two points are (-6,3) and (6,-3).<BR>\n" );
document.write( "Now the perpendicular line that would be the shortest distance to the original line has a slope that is the negative reciprocal of the original line.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=2%2F3 ALIGN=MIDDLE  ALT=\"m=2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You have the slope and the point, find the line.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-3=%282%2F3%29%28x-%28-6%29%29 ALIGN=MIDDLE  ALT=\"y-3=%282%2F3%29%28x-%28-6%29%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=%282%2F3%29%28x%2B6%29 ALIGN=MIDDLE  ALT=\"y-3=%282%2F3%29%28x%2B6%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=%282%2F3%29x%2B4 ALIGN=MIDDLE  ALT=\"y-3=%282%2F3%29x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F3%29x%2B7 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29x%2B7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-10%2C2%2C-2%2C10%2Cgrid%281%29%2Ccircle%28-6%2C3%2C0.3%29%2Ccircle%286%2C-3%2C0.3%29%2Cgraph%28300%2C300%2C-10%2C2%2C-2%2C10%2C-%283%2F2%29x-1%2F2%2C-sqrt%2818%28x%5E2%2F24-1%29%29%2Csqrt%2818%28x%5E2%2F24-1%29%29%2C%282%2F3%29x%2B7%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now find the intersection point of the two lines to find the other point to use to calculate minimum distance.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%283%2F2%29x-1%2F2 ALIGN=MIDDLE  ALT=\"y=-%283%2F2%29x-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F3%29x%2B7 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29x%2B7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%283%2F2%29x-1%2F2=%282%2F3%29x%2B7 ALIGN=MIDDLE  ALT=\"-%283%2F2%29x-1%2F2=%282%2F3%29x%2B7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Multiply both side by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9x-3=4x%2B42 ALIGN=MIDDLE  ALT=\"-9x-3=4x%2B42\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-13x=45 ALIGN=MIDDLE  ALT=\"-13x=45\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=-45%2F13%29 ALIGN=MIDDLE  ALT=\"highlight%28x=-45%2F13%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F3%29%28-45%2F13%29%2B7 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29%28-45%2F13%29%2B7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%2830%2F13%29%2B91%2F13 ALIGN=MIDDLE  ALT=\"y=-%2830%2F13%29%2B91%2F13\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=61%2F13%29 ALIGN=MIDDLE  ALT=\"highlight%28y=61%2F13%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So now use the distance formula,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5E2=%28-6-%28-45%2F13%29%29%5E2%2B%283-61%2F13%29%5E2 ALIGN=MIDDLE  ALT=\"D%5E2=%28-6-%28-45%2F13%29%29%5E2%2B%283-61%2F13%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5E2=%28-78%2F13%2B45%2F13%29%5E2%2B%2839%2F13-61%2F13%29%5E2 ALIGN=MIDDLE  ALT=\"D%5E2=%28-78%2F13%2B45%2F13%29%5E2%2B%2839%2F13-61%2F13%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5E2=%28-33%2F13%29%5E2%2B%2822%2F13%29%5E2 ALIGN=MIDDLE  ALT=\"D%5E2=%28-33%2F13%29%5E2%2B%2822%2F13%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5E2=%281089%2F169%29%2B%28484%2F169%29 ALIGN=MIDDLE  ALT=\"D%5E2=%281089%2F169%29%2B%28484%2F169%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5E2=1573%2F169 ALIGN=MIDDLE  ALT=\"D%5E2=1573%2F169\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=sqrt%281573%29%2F13 ALIGN=MIDDLE  ALT=\"D=sqrt%281573%29%2F13\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28D=%2811%2F13%29sqrt%2813%29%29 ALIGN=MIDDLE  ALT=\"highlight%28D=%2811%2F13%29sqrt%2813%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( ".\r<BR>\n" );
document.write( "\n" );
document.write( "You can also calculate the other point that intersects with (6,-3) and then calculate the distance. <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );