document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1025651>Question 1025651</A>: <I><SPAN STYLE=\"word-break: break-all;\"> if the lines x-py+ap^2=0<BR>\n" );
document.write( "Ad x-qy+aq^2 are perpendicular <BR>\n" );
document.write( "Show that pq=-1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #640957 by </B><A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><B>josgarithmetic(39617)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josgarithmetic><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=640957\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The shortcut to proving it is to know what to recognize in the standard form equation for a line.  The slopes of your given lines in order are  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fp ALIGN=MIDDLE  ALT=\"1%2Fp\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fq ALIGN=MIDDLE  ALT=\"1%2Fq\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  The lines to be perpendicular will require  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Fp%29%281%2Fp%29=-1 ALIGN=MIDDLE  ALT=\"%281%2Fp%29%281%2Fp%29=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Multiply left and right members by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pq ALIGN=MIDDLE  ALT=\"pq\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  <BR>\n" );
document.write( ".... there, you have it.\r<BR>\n" );
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document.write( "So you want a little more information for this.<BR>\n" );
document.write( "Basic fact is,  if product of the slopes of two lines is negative ONE, then the two lines are perpendicular.\r<BR>\n" );
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document.write( "Put each equation into slope-intercept form.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-py%2Bap%5E2=0 ALIGN=MIDDLE  ALT=\"x-py%2Bap%5E2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-py=-x-ap%5E2 ALIGN=MIDDLE  ALT=\"-py=-x-ap%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2Fp%29x%2Bap%5E2 ALIGN=MIDDLE  ALT=\"y=%281%2Fp%29x%2Bap%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-qy%2Baq%5E2=0 ALIGN=MIDDLE  ALT=\"x-qy%2Baq%5E2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-qy=-x-aq%5E2 ALIGN=MIDDLE  ALT=\"-qy=-x-aq%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2Fq%29x%2Baq%5E2 ALIGN=MIDDLE  ALT=\"y=%281%2Fq%29x%2Baq%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Note the slopes of the two lines are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fp ALIGN=MIDDLE  ALT=\"1%2Fp\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fq ALIGN=MIDDLE  ALT=\"1%2Fq\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The two lines are GIVEN as perpendicular, and this means according to the basic fact about perpendicular lines in the plane,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Fp%29%281%2Fq%29=-1 ALIGN=MIDDLE  ALT=\"%281%2Fp%29%281%2Fq%29=-1\"   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=%281%2Fp%29%281%2Fq%29%2Apq=-1%2A%28pq%29 ALIGN=MIDDLE  ALT=\"%281%2Fp%29%281%2Fq%29%2Apq=-1%2A%28pq%29\"   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=1=-1%2Apq ALIGN=MIDDLE  ALT=\"1=-1%2Apq\"   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=1%2A%28-1%29=-1%2Apq%2A%28-1%29 ALIGN=MIDDLE  ALT=\"1%2A%28-1%29=-1%2Apq%2A%28-1%29\"   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=-1=pq ALIGN=MIDDLE  ALT=\"-1=pq\"   DISABLED_style= \"max-width:90%; height:auto;\" >-------done. </SPAN>\n" );
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