document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1128589>Question 1128589</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The point A lies on the parabola y=x^2 - 6x +24. The point B lies on the parabola y= 6x-x^2 and is vertically below the point A.<BR>\n" );
document.write( "a) Show this information on a neat diagram<BR>\n" );
document.write( "b) Find the least value of the length of line AB </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #745122 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=745122\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The 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;\" > lies on the parabola <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2+-+6x+%2B24 ALIGN=MIDDLE  ALT=\"y=x%5E2+-+6x+%2B24\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "The 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;\" > lies on the parabola <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=+6x-x%5E2 ALIGN=MIDDLE  ALT=\"y=+6x-x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=vertically ALIGN=MIDDLE  ALT=\"vertically\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+below+ ALIGN=MIDDLE  ALT=\"+below+\"   DISABLED_style= \"max-width:90%; height:auto;\" >the 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;\" >.\r<BR>\n" );
document.write( "\n" );
document.write( "solution:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2+-+6x+%2B24 ALIGN=MIDDLE  ALT=\"y=x%5E2+-+6x+%2B24\"   DISABLED_style= \"max-width:90%; height:auto;\" > -> parabola is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=opening ALIGN=MIDDLE  ALT=\"opening\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+up ALIGN=MIDDLE  ALT=\"+up\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the 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;\" > in  vertex (minimum)\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=+-x%5E2%2B6x ALIGN=MIDDLE  ALT=\"y=+-x%5E2%2B6x\"   DISABLED_style= \"max-width:90%; height:auto;\" > -> parabola is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=opening ALIGN=MIDDLE  ALT=\"opening\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+down ALIGN=MIDDLE  ALT=\"+down\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the 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;\" > in  vertex (maximum)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "writ both equations in vertex form:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28x%5E2+-+6x%29+%2B24 ALIGN=MIDDLE  ALT=\"y=%28x%5E2+-+6x%29+%2B24\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28x%5E2+-+6x%2Bb%5E2%29+-b%5E2%2B24 ALIGN=MIDDLE  ALT=\"y=%28x%5E2+-+6x%2Bb%5E2%29+-b%5E2%2B24\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28x%5E2+-+6x%2B3%5E2%29+-3%5E2%2B24 ALIGN=MIDDLE  ALT=\"y=%28x%5E2+-+6x%2B3%5E2%29+-3%5E2%2B24\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28x+-+3%29%5E2+-9%2B24 ALIGN=MIDDLE  ALT=\"y=%28x+-+3%29%5E2+-9%2B24\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28x+-+3%29%5E2+%2B15 ALIGN=MIDDLE  ALT=\"y=%28x+-+3%29%5E2+%2B15\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "=> vertex is at (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15 ALIGN=MIDDLE  ALT=\"15\"   DISABLED_style= \"max-width:90%; height:auto;\" >)=>the 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;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=+-x%5E2%2B6x ALIGN=MIDDLE  ALT=\"y=+-x%5E2%2B6x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=+-%28x%5E2-6x%2Bb%5E2%29-1%28-b%5E2%29 ALIGN=MIDDLE  ALT=\"y=+-%28x%5E2-6x%2Bb%5E2%29-1%28-b%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=+-%28x%5E2-6x%2B3%5E2%29-1%28-3%5E2%29 ALIGN=MIDDLE  ALT=\"y=+-%28x%5E2-6x%2B3%5E2%29-1%28-3%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=+-%28x-3%29%5E2%2B9 ALIGN=MIDDLE  ALT=\"y=+-%28x-3%29%5E2%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "=> vertex is at (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" >)=>the 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;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "since both points have same <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > coordinate, the  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;\" > lies vertically below the 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;\" > because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > coordinate of the 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;\" > is greater than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > coordinate of the 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;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "a) Show this information on a neat diagram\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing+%28+600%2C+600%2C+-20%2C+20%2C+-20%2C+20%2C%0D%0Acircle%283%2C15%2C.12%29%2Ccircle%283%2C9%2C.12%29%2C%0D%0A%0D%0Alocate%283%2C15%2CA%283%2C15%29%29%2C+locate%283%2C9%2CB%283%2C9%29%29%2C%0D%0A%0D%0Aline%283%2C15%2C3%2C9%29%2C%0D%0A%0D%0Agraph%28+600%2C+600%2C+-20%2C+20%2C+-20%2C+20%2C+%28x+-+3%29%5E2+%2B15%2C+-%28x-3%29%5E2%2B9%29%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "b) Find the least value of the length of line AB \r<BR>\n" );
document.write( "\n" );
document.write( "the length of line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB=15-9=6 ALIGN=MIDDLE  ALT=\"AB=15-9=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );