document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=125671>Question 125671</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the minimum distance between f(x)=x^2 and y=2x-5 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #92044 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=92044\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let's first graph given parabola <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2 ALIGN=MIDDLE  ALT=\"y=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and a line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x-5 ALIGN=MIDDLE  ALT=\"y=2x-5\"   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=graph%28+600%2C+400%2C+-10%2C+10%2C+-20%2C+20%2C+x%5E2%2C+2x-5%29+ ALIGN=MIDDLE  ALT=\"graph%28+600%2C+400%2C+-10%2C+10%2C+-20%2C+20%2C+x%5E2%2C+2x-5%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "as you can see, there are no points of intersection\r<BR>\n" );
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document.write( "on a graph above we have given parabola <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2 ALIGN=MIDDLE  ALT=\"y=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-5 ALIGN=MIDDLE  ALT=\"2x-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "now, I will add a line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-1 ALIGN=MIDDLE  ALT=\"2x-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > parallel to the given line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-5 ALIGN=MIDDLE  ALT=\"2x-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >(these lines have same slopes) which will be a tangent (there will be one point of intersection) on parabola <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >; the distance between these two parallel lines will be equal to the distance between the given line and parabola\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+600%2C+400%2C+-10%2C+10%2C+-20%2C+20%2C+x%5E2%2C+2x-5%2C+2x-1%29+ ALIGN=MIDDLE  ALT=\"graph%28+600%2C+400%2C+-10%2C+10%2C+-20%2C+20%2C+x%5E2%2C+2x-5%2C+2x-1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "now, we can choose two points (one on each of parallel lines) and find a distance between them; it will be also equal to the distance between parabola and the given line\r<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=hummingbird_distance_2D.solver>Distance between two points in two dimensions</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>The distance (denoted by d) between two points in two dimensions is given by the following formula:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=sqrt%28%28x1-x2%29%5E2+%2B+%28y1-y2%29%5E2%29 ALIGN=MIDDLE  ALT=\"d=sqrt%28%28x1-x2%29%5E2+%2B+%28y1-y2%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Thus in our case, the required distance is<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=sqrt%28%280-0%29%5E2+%2B+%28-1--5%29%5E2%29=+4+ ALIGN=MIDDLE  ALT=\"d=sqrt%28%280-0%29%5E2+%2B+%28-1--5%29%5E2%29=+4+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  For more on this concept, refer to <A  rel=nofollow HREF=Distance_formula.wikipedia>Distance formula</A>.<BR>\n" );
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document.write( "answer is: the distance is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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