document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1184304>Question 1184304</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The points representing the complex number z for which the arg<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28z+-+2%29%2F%28z+%2B+2%29+=+pi%2F3+ ALIGN=MIDDLE  ALT=\"+%28z+-+2%29%2F%28z+%2B+2%29+=+pi%2F3+\"   DISABLED_style= \"max-width:90%; height:auto;\" > lie on which of the following:<BR>\n" );
document.write( "A- Circle<BR>\n" );
document.write( "B- Straight Line<BR>\n" );
document.write( "C- Ellipse<BR>\n" );
document.write( "D- Parabola<BR>\n" );
document.write( "[Note: Here arg means argument and pi means the value 3.142...] </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #814900 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=814900\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ===> Let z be a complex number satisfying the given condition.\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28z-2%29%2F%28z%2B2%29+=+%28%28z-2%29%2F%28z%2B2%29%29%2A%28%28%28z%2B2%29%29%5Bc%5D%2F%28%28z%2B2%29%29%5Bc%5D%29+=+%28%28z-2%29%2F%28z%2B2%29%29%2A%28%28z%5Bc%5D%2B2%29%2F%28z%5Bc%5D%2B2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >,\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "where the subscript \"c\" denotes conjugation.\r<BR>\n" );
document.write( "\n" );
document.write( "===><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28z%2Az%5Bc%5D+%2B+2z+-+2z%5Bc%5D+-+4%29%2F%28z%2Az%5Bc%5D+%2B+2z+%2B+2z%5Bc%5D+%2B+4%29+=+%28abs%28z%29%5E2+%2B+4%2AIm%28z%29%2Ai-4%29%2F%28abs%28z%29%5E2+%2B+4%2ARe%28z%29+%2B+4%29+=+%28abs%28z%29%5E2-4%29%2F%28abs%28z%29%5E2+%2B+4%2ARe%28z%29+%2B+4%29+%2B+%28%284%2AIm%28z%29%29%2F+%28abs%28z%29%5E2+%2B+4%2ARe%28z%29+%2B+4%29%29%2Ai%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha+%2B+i%2A%28beta%29 ALIGN=MIDDLE  ALT=\"alpha+%2B+i%2A%28beta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.   \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The given condition then says that \r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28theta%29+=+tan%28pi%2F3%29+=+sqrt%283%29=+beta%2Falpha+=+%284%2AIm%28z%29%29%2F%28abs%28z%29%5E2+-+4%29 ALIGN=MIDDLE   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=sqrt%283%29=+%284%2AIm%28z%29%29%2F%28abs%28z%29%5E2+-4%29 ALIGN=MIDDLE  ALT=\"sqrt%283%29=+%284%2AIm%28z%29%29%2F%28abs%28z%29%5E2+-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  assuming that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28z%29 ALIGN=MIDDLE  ALT=\"abs%28z%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > doesn't lie on the circle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28z%29+=+2 ALIGN=MIDDLE  ALT=\"abs%28z%29+=+2\"   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=abs%28z%29%5E2+=+%284%2Fsqrt%283%29%29%2AIm%28z%29%2B4 ALIGN=MIDDLE  ALT=\"abs%28z%29%5E2+=+%284%2Fsqrt%283%29%29%2AIm%28z%29%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  after solving for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28z%29%5E2 ALIGN=MIDDLE  ALT=\"abs%28z%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If we let z = x + i*y, then the preceding equation says that \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28x%5E2+%2B+y%5E2+=+%284%2Fsqrt%283%29%29%2Ay+%2B+4%29 ALIGN=MIDDLE  ALT=\"red%28x%5E2+%2B+y%5E2+=+%284%2Fsqrt%283%29%29%2Ay+%2B+4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <----this circle!\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "which is clearly an equation of a circle.  Therefore the answer is option <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28A%29 ALIGN=MIDDLE  ALT=\"highlight%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  \r<BR>\n" );
document.write( "\n" );
document.write( "Whether anyone denies this or not is immaterial, because the z values satisfying <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+arg%28%28z+-+2%29%2F%28z+%2B+2%29%29+=+pi%2F3+ ALIGN=MIDDLE  ALT=\"+arg%28%28z+-+2%29%2F%28z+%2B+2%29%29+=+pi%2F3+\"   DISABLED_style= \"max-width:90%; height:auto;\" > still <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28LIES%29 ALIGN=MIDDLE  ALT=\"red%28LIES%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28ON%29 ALIGN=MIDDLE  ALT=\"red%28ON%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > this circle.  <BR>\n" );
document.write( "It doesn't lie on a straight line, not on an ellipse, not ona parabola, but on a CIRCLE.  \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "And even if z is only on a small part of the circle doesn't matter, because IT IS STILL ON THE CIRCLE. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );