document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=17683>Question 17683</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi i would really appreciate it if someone would help me with this problem.\r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%2A%28q-r%29%2Ax%5E2%2Bq%2A%28r-p%29%2Ax%2Ay%2Br%2A%28p-q%29%2Ay%5E2 ALIGN=MIDDLE  ALT=\"p%2A%28q-r%29%2Ax%5E2%2Bq%2A%28r-p%29%2Ax%2Ay%2Br%2A%28p-q%29%2Ay%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a perfect square, then prove p,q,r are in harmonic progression.\r<BR>\n" );
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document.write( "I know that i should prove somehow that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q=%282%2Ap%2Ar%29%2F%28p%2Br%29 ALIGN=MIDDLE  ALT=\"q=%282%2Ap%2Ar%29%2F%28p%2Br%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > i think i can do this using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2=4%2Aa%2Ac ALIGN=MIDDLE  ALT=\"b%5E2=4%2Aa%2Ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > but i thought that it only applied to quadratic equations of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Ax%5E2%2Bb%2Ax%2Bc ALIGN=MIDDLE  ALT=\"a%2Ax%5E2%2Bb%2Ax%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > Please help. Thanks </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #8540 by </B><A HREF=/tutors/aboutme.mpl?userid=venugopalramana><B>venugopalramana(3286)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=venugopalramana><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.geocities.com/venugopalramana/FREE_MATHS_ASSISTANCE.html valign=middle><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=8540\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%2A%28q-r%29%2Ax%5E2%2Bq%2A%28r-p%29%2Ax%2Ay%2Br%2A%28p-q%29%2Ay%5E2 ALIGN=MIDDLE  ALT=\"p%2A%28q-r%29%2Ax%5E2%2Bq%2A%28r-p%29%2Ax%2Ay%2Br%2A%28p-q%29%2Ay%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a perfect square, then prove p,q,r are in harmonic progression.\r<BR>\n" );
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document.write( "I know that i should prove somehow that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q=%282%2Ap%2Ar%29%2F%28p%2Br%29 ALIGN=MIDDLE  ALT=\"q=%282%2Ap%2Ar%29%2F%28p%2Br%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > i think i can do this using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2=4%2Aa%2Ac ALIGN=MIDDLE  ALT=\"b%5E2=4%2Aa%2Ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > but i thought that it only applied to quadratic equations of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Ax%5E2%2Bb%2Ax%2Bc ALIGN=MIDDLE  ALT=\"a%2Ax%5E2%2Bb%2Ax%2Bc\"   DISABLED_style= \"max-width:90%; height:auto;\" > Please help. Thanks<BR>\n" );
document.write( "YOU ARE PERFECTLY CORRECT...PROCEED AND YOU WOULD HAVE GOT THE ANSWER BY YOUR SELF.DONT WORRY ABOUT PRESENCE OF CONSTANTS LIKE A,B,C OR VARIABLES LIKE Y,Y^2 ETC.IN THEIR PLACE..IT IS ALL SAME AS LONG AS YOU HAVE A QUADRATIC IN ONE VARIABLE.IT COULD BE IN X OR Y.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2=4%2Aa%2Ac ALIGN=MIDDLE  ALT=\"b%5E2=4%2Aa%2Ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > =0 for the given equation to be a perfect square..we have <BR>\n" );
document.write( "a=p(q-r)..b=q(r-p)y...c=r*(p-q)*y^2..so,we have<BR>\n" );
document.write( "b^2=4ac ...hence...<BR>\n" );
document.write( "q^2*(r-p)^2*y^2=4*p*(q-r)*r*(p-q)*y^2<BR>\n" );
document.write( "q^2*(r-p)^2=4*p*(q-r)*r*(p-q)<BR>\n" );
document.write( "..NOW HERE LET US USE A SMALL TRICK TO SIMPLIFY THINGS OR DO IT ORALLY....<BR>\n" );
document.write( "LET t=p(q-r)...u=q(r-p)....v=r(p-q).so that.t+u+v=0.Or.-u=(t+v)..or.u^2=(t+v)^2<BR>\n" );
document.write( "but u^2=4tv from above q^2*(r-p)^2=4*p*(q-r)*r*(p-q)<BR>\n" );
document.write( "hence (t+v)^2=4tv..or (t-v)^2=0 ..or t=v..or p(q-r)=r(p-q).<BR>\n" );
document.write( "NOW YOU HAVE YOUR DESIRED EQUATION <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q=%282%2Ap%2Ar%29%2F%28p%2Br%29 ALIGN=MIDDLE  ALT=\"q=%282%2Ap%2Ar%29%2F%28p%2Br%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and hence p,q,r are in H.P. </SPAN>\n" );
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