document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1032691>Question 1032691</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this;<BR>\n" );
document.write( "Given are two points A(-1,3) and B(3,9).(a) Show that C(5,12)is a point of AB. (b) A Point P(x,y) moves in such a way that AP^2+CP^2=2BP^2. Find the equation of the locus of P. (c)Show that this locus is a straight line perpendicular to AB.\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #647336 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=647336\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> (a) is easy.<BR>\n" );
document.write( "A picture is not needed, but I will add a picture so you can visualize it:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C600%2C-2%2C8%2C-2%2C14%2C%0D%0Agrid%280%29%2Ccircle%28-1%2C3%2C0.1%29%2C%0D%0Acircle%283%2C9%2C0.1%29%2Ccircle%285%2C12%2C0.1%29%2C%0D%0Agreen%28triangle%28-1%2C3%2C3%2C3%2C3%2C9%29%29%2C%0D%0Ared%28triangle%283%2C9%2C5%2C9%2C5%2C12%29%29%2C%0D%0Alocate%28-1.5%2C3.2%2CA%29%2Clocate%282.5%2C9.3%2CB%29%2Clocate%284.9%2C12.5%2CC%29%2C%0D%0Alocate%280.3%2C3.3%2Cgreen%28x%5BB%5D-x%5BA%5D%29%29%2Clocate%283.1%2C6.2%2Cgreen%28y%5BB%5D-y%5BA%5D%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Given points <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28x%5BA%5D%2Cy%5BA%5D%29 ALIGN=MIDDLE  ALT=\"A%28x%5BA%5D%2Cy%5BA%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%28x%5BB%5D%2Cy%5BB%5D%29 ALIGN=MIDDLE  ALT=\"B%28x%5BB%5D%2Cy%5BB%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the slope of AB is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y%5BB%5D-y%5BA%5D%29%2F%28x%5BB%5D-x%5BA%5D%29 ALIGN=MIDDLE  ALT=\"%28y%5BB%5D-y%5BA%5D%29%2F%28x%5BB%5D-x%5BA%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28-1%2C3%29 ALIGN=MIDDLE  ALT=\"A%28-1%2C3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%283%2C9%29 ALIGN=MIDDLE  ALT=\"B%283%2C9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the slope of AB is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%289-3%29%2F%283-%28-1%29%29=6%2F%283%2B1%29=6%2F4=3%2F2 ALIGN=MIDDLE  ALT=\"%289-3%29%2F%283-%28-1%29%29=6%2F%283%2B1%29=6%2F4=3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%283%2C9%29 ALIGN=MIDDLE  ALT=\"B%283%2C9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%285%2C12%29 ALIGN=MIDDLE  ALT=\"C%285%2C12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the slope of BC is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2812-9%29%2F%285-3%29=3%2F2 ALIGN=MIDDLE  ALT=\"%2812-9%29%2F%285-3%29=3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Since AB and BC have the same slope, they are either parallel or the same line.<BR>\n" );
document.write( "As they have point B in common, AB and BC are the same line.<BR>\n" );
document.write( " <BR>\n" );
document.write( "(b) ONE WAY TO GO ABOUT IT (ugly, but probably the expected way):<BR>\n" );
document.write( "If point P is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%2Cy%29 ALIGN=MIDDLE  ALT=\"P%28x%2Cy%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AP%5E2=%28x-%28-1%29%29%5E2%2B%28y-3%29%5E2=%28x%2B1%29%5E2%2By%5E2-6y%2B9=x%5E2%2B2x%2B1%2By%5E2-6y%2B9=x%5E2%2By%5E2%2B2x-6y%2B10 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BP%5E2=%28x-3%29%5E2%2B%28y-9%29%5E2=x%5E2-6x%2B9%2By%5E2-18y%2B81=x%5E2%2By%5E2-6x-18y%2B90 ALIGN=MIDDLE  ALT=\"BP%5E2=%28x-3%29%5E2%2B%28y-9%29%5E2=x%5E2-6x%2B9%2By%5E2-18y%2B81=x%5E2%2By%5E2-6x-18y%2B90\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CP%5E2=%28x-5%29%5E2%2B%28y-12%29%5E2=x%5E2-10x%2B25%2By%5E2-24y%2B144=x%5E2%2By%5E2-10x-24y%2B169 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AP%5E2%2BCP%5E2=x%5E2%2By%5E2%2B2x-6y%2B10%2Bx%5E2%2By%5E2-10x-24y%2B169=2x%5E2%2B2y%5E2-8x-30y%2B179 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2bP%5E2=2%28x%5E2%2By%5E2-6x-18y%2B90%29=2x%5E2%2B2y%5E2-12x-36y%2B180 ALIGN=MIDDLE  ALT=\"2bP%5E2=2%28x%5E2%2By%5E2-6x-18y%2B90%29=2x%5E2%2B2y%5E2-12x-36y%2B180\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AP%5E2%2BCP%5E2=2BP%5E2 ALIGN=MIDDLE  ALT=\"AP%5E2%2BCP%5E2=2BP%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > means<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2B2y%5E2-8x-30y%2B179=2x%5E2%2B2y%5E2-12x-36y%2B180 ALIGN=MIDDLE  ALT=\"2x%5E2%2B2y%5E2-8x-30y%2B179=2x%5E2%2B2y%5E2-12x-36y%2B180\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8x-30y%2B179=-12x-36y%2B180 ALIGN=MIDDLE  ALT=\"-8x-30y%2B179=-12x-36y%2B180\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x-8x%2B36y-30y=180-179 ALIGN=MIDDLE  ALT=\"12x-8x%2B36y-30y=180-179\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%2B6y=1 ALIGN=MIDDLE  ALT=\"4x%2B6y=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "That is the equation of a straight line, which is the locus of P.<BR>\n" );
document.write( "Transforming the equation into slope-intercept form, we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%2B6y=1 ALIGN=MIDDLE  ALT=\"4x%2B6y=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6y=-4x%2B1 ALIGN=MIDDLE  ALT=\"6y=-4x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-4x%2B1%29%2F6 ALIGN=MIDDLE  ALT=\"y=%28-4x%2B1%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-2%2F3x%29%2B1%2F6 ALIGN=MIDDLE  ALT=\"y=%28-2%2F3x%29%2B1%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We could also just find the slope, using a formula.<BR>\n" );
document.write( "Either way, the slope of the line is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%2F3 ALIGN=MIDDLE  ALT=\"-2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If the product of that slope and the slope of AB (found in part (a) is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "then the lines are perpendicular, and it so happens that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2%2F3%29%283%2F2%29=-1 ALIGN=MIDDLE  ALT=\"%28-2%2F3%29%283%2F2%29=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so the locus of P is a line perpendicular to AB.<BR>\n" );
document.write( " <BR>\n" );
document.write( "ANOTHER WAY (possible, depending on what you have already covered in math classes):<BR>\n" );
document.write( "When calculating the slopes of AB and BC,<BR>\n" );
document.write( "you may have noticed that for points A, B and C<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5BB%5D-x%5BA%5D=2%28x%5BC%5D-x%5BB%5D%29 ALIGN=MIDDLE  ALT=\"x%5BB%5D-x%5BA%5D=2%28x%5BC%5D-x%5BB%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5BB%5D-y%5BA%5D=2%28y%5BC%5D-y%5BB%5D%29 ALIGN=MIDDLE  ALT=\"y%5BB%5D-y%5BA%5D=2%28y%5BC%5D-y%5BB%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That tells you that for the distances <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB=2BC ALIGN=MIDDLE  ALT=\"AB=2BC\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We could calculate those distances, but I only care about their ratios,<BR>\n" );
document.write( "so for easier writing, I will rename the distances as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BC=c ALIGN=MIDDLE  ALT=\"BC=c\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB=2c ALIGN=MIDDLE  ALT=\"AB=2c\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "You may think of point P as not being on line AB.<BR>\n" );
document.write( "However, as it is a moving point, at some point it could be on line AB,<BR>\n" );
document.write( "but in that very special case, I would call it point D, and I will say it is a distance <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the other side of A.<BR>\n" );
document.write( "I will find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The situation would be like this, with the distances:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28800%2C100%2C-1.5%2C6.5%2C-0.5%2C0.5%2C%0D%0Aarrow%28-2%2C0%2C9%2C0%29%2Ccircle%28-1%2C0%2C0.05%29%2Clocate%28-1.1%2C0.2%2CD%29%2C%0D%0Acircle%280%2C0%2C0.05%29%2Clocate%28-0.05%2C0.2%2CA%29%2C%0D%0Acircle%284%2C0%2C0.05%29%2Clocate%283.95%2C0.2%2CB%29%2C%0D%0Acircle%286%2C0%2C0.05%29%2Clocate%286.05%2C0.2%2CC%29%2C%0D%0Alocate%28-0.55%2C0%2Cd%29%2Clocate%281.9%2C0%2C2c%29%2Clocate%284.95%2C0%2Cc%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > . (If D is not on the side of A I assumed it to be, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > will be a negative value).<BR>\n" );
document.write( "Since now P is at D, the equation with the squares is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5E2%2B%283c%2Bd%29%5E2=2%282c%2Bd%29%5E2 ALIGN=MIDDLE  ALT=\"d%5E2%2B%283c%2Bd%29%5E2=2%282c%2Bd%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%2B9c%5E2%2B6cd%2Bd%5E2=2%284c%5E2%2B4cd%2Bd%5E2%29 ALIGN=MIDDLE  ALT=\"d%5E2%2B9c%5E2%2B6cd%2Bd%5E2=2%284c%5E2%2B4cd%2Bd%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2d%5E2%2B9c%5E2%2B6cd=8c%5E2%2B8cd%2B2d%5E2 ALIGN=MIDDLE  ALT=\"2d%5E2%2B9c%5E2%2B6cd=8c%5E2%2B8cd%2B2d%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9c%5E2%2B6cd=8c%5E2%2B8cd%7D%7D%0D%0A%7B%7B%7B9c%5E2-8c%5E2=8cd-6cd ALIGN=MIDDLE  ALT=\"9c%5E2%2B6cd=8c%5E2%2B8cd%7D%7D%0D%0A%7B%7B%7B9c%5E2-8c%5E2=8cd-6cd\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2=2cd ALIGN=MIDDLE  ALT=\"c%5E2=2cd\"   DISABLED_style= \"max-width:90%; height:auto;\" > and since we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%3C%3E0 ALIGN=MIDDLE  ALT=\"c%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (even though we did not calculate it), we divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" > and get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=2d ALIGN=MIDDLE  ALT=\"c=2d\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Now, what about a point P that is not D?<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28800%2C500%2C-1.5%2C6.5%2C-0.5%2C4.5%2C%0D%0Aarrow%28-2%2C0%2C9%2C0%29%2Ccircle%28-1%2C0%2C0.05%29%2Clocate%28-1.1%2C0.2%2CD%29%2C%0D%0Acircle%28-0.8%2C4%2C0.05%29%2Clocate%28-0.85%2C4.2%2CP%29%2Ctriangle%28-1%2C0%2C-0.8%2C4%2C0%2C0%29%2C%0D%0Acircle%280%2C0%2C0.05%29%2Clocate%280%2C0.2%2CA%29%2Ctriangle%284%2C0%2C-0.8%2C4%2C6%2C0%29%2C%0D%0Acircle%284%2C0%2C0.05%29%2Clocate%283.95%2C0.2%2CB%29%2C%0D%0Acircle%286%2C0%2C0.05%29%2Clocate%286.05%2C0.2%2CC%29%2C%0D%0Alocate%28-0.55%2C0%2Cd%29%2Clocate%281.9%2C0%2C4d%29%2Clocate%284.95%2C0%2C2d%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "With the Pythagorean theorem (applied to all 3 triangles including side DP), we can easily prove that if P is such that AB and DP are perpendicular, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AP%5E2%2BCP%5E2=2BP%5E2 ALIGN=MIDDLE  ALT=\"AP%5E2%2BCP%5E2=2BP%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The other way around (proving that if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AP%5E2%2BCP%5E2=2BP%5E2 ALIGN=MIDDLE  ALT=\"AP%5E2%2BCP%5E2=2BP%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , then AP and DP are perpendicular) I believe requires using the law of cosines, which may be beyond what you have studied.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );