document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1055178>Question 1055178</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the minimum distance between the 2 parabolas:<BR>\n" );
document.write( "y = x^2+4 and x = y^2 ? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #670381 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=670381\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> OK, assume that the shortest distance line segment intersects the first curve at (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2B4 ALIGN=MIDDLE  ALT=\"a%5E2%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and intersects the second curve at (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2 ALIGN=MIDDLE  ALT=\"b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" >).<BR>\n" );
document.write( "The slope of the tangent line of the first curve at point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a ALIGN=MIDDLE  ALT=\"2a\"   DISABLED_style= \"max-width:90%; height:auto;\" > since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx=2x ALIGN=MIDDLE  ALT=\"dy%2Fdx=2x\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The slope of the tangent line of the second curve at point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%282b%29 ALIGN=MIDDLE  ALT=\"1%2F%282b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx=1%2F%282%2Asqrt%28x%29%29 ALIGN=MIDDLE  ALT=\"dy%2Fdx=1%2F%282%2Asqrt%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The slopes are equivalent which leads to the first equation,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a=1%2F%282b%29 ALIGN=MIDDLE  ALT=\"2a=1%2F%282b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4ab=1 ALIGN=MIDDLE  ALT=\"4ab=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now use the distance formula (actually use the distance squared to eliminate the square root) with the two points of intersection,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5E2=%28a-b%5E2%29%5E2%2B%28a%5E2%2B4-b%29%5E2 ALIGN=MIDDLE  ALT=\"D%5E2=%28a-b%5E2%29%5E2%2B%28a%5E2%2B4-b%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "From the previous equation,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1%2F%284b%29 ALIGN=MIDDLE  ALT=\"a=1%2F%284b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substitute,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5E2=%281%2F%284b%29-b%5E2%29%5E2%2B%281%2F%2816b%29%2B4-b%29%5E2 ALIGN=MIDDLE  ALT=\"D%5E2=%281%2F%284b%29-b%5E2%29%5E2%2B%281%2F%2816b%29%2B4-b%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Graphing and finding the minimum,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=1.188 ALIGN=MIDDLE  ALT=\"b=1.188\"   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=a=1%2F%284b%29 ALIGN=MIDDLE  ALT=\"a=1%2F%284b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=0.210438 ALIGN=MIDDLE  ALT=\"a=0.210438\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/display-illustration.mpl?tutor=Fombitz&illustration_name=lp2.JPG\">.<BR>\n" );
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document.write( "So then the points are (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.210 ALIGN=MIDDLE  ALT=\"0.210\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4.044 ALIGN=MIDDLE  ALT=\"4.044\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.188 ALIGN=MIDDLE  ALT=\"1.188\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.411 ALIGN=MIDDLE  ALT=\"1.411\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "Using the distance formula,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5Bmin%5D=3.098 ALIGN=MIDDLE  ALT=\"D%5Bmin%5D=3.098\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You could also have just used the graphed value when we minimized the distance squared,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5Bmin%5D%5E2=9.601 ALIGN=MIDDLE  ALT=\"D%5Bmin%5D%5E2=9.601\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5Bmin%5D=3.098 ALIGN=MIDDLE  ALT=\"D%5Bmin%5D=3.098\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/display-illustration.mpl?tutor=Fombitz&illustration_name=lp3.JPG\">. </SPAN>\n" );
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