document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1099803>Question 1099803</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If z-2i/2z-1 is purely imaginary prove that the locus of z in the Argands diagram is a circle .find centre and radius  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #714398 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13203)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=714398\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br>This is a curious type of problem that I have not seen before....<br><BR>\n" );
document.write( "Definitely try to get a second opinion....<br><BR>\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+a%2Bbi ALIGN=MIDDLE  ALT=\"z+=+a%2Bbi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28z-2i%29%2F%282z-1%29+=+%28a%2B%28b-2%29i%29%2F%28%282a-1%29%2B%282b%29i%29 ALIGN=MIDDLE  ALT=\"%28z-2i%29%2F%282z-1%29+=+%28a%2B%28b-2%29i%29%2F%28%282a-1%29%2B%282b%29i%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Rationalize the denominator by multiplying by its conjugate:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28a%2B%28b-2%29i%29%28%282a-1%29-%282b%29i%29%29%2F%28%28%282a-1%29%2B%282b%29i%29%28%282a-1%29-%282b%29i%29%29+=+%28%282a%5E2-2%29%2B%28-2abi%29%2B%28%28b-2%29%282a-1%29i%29%2B%282b%5E2-4b%29%29%2F%28%282a-1%29%2B%282b%29i%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The condition for that number to be purely imaginary is the the real part must be 0:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%5E2-a%2B2b%5E2-4b+=+0 ALIGN=MIDDLE  ALT=\"2a%5E2-a%2B2b%5E2-4b+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "With the a^2 and b^2 terms with the same coefficient, that is indeed the equation of a circle.<br><BR>\n" );
document.write( "To find the center and radius of the circle, complete the squares in a and b:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28a%5E2-%281%2F2%29a%2B1%2F16%29+%2B+2%28b%5E2-2b%2B1%29+=+0%2B2%2F16%2B2+=+17%2F8 ALIGN=MIDDLE  ALT=\"2%28a%5E2-%281%2F2%29a%2B1%2F16%29+%2B+2%28b%5E2-2b%2B1%29+=+0%2B2%2F16%2B2+=+17%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a-%281%2F4%29%29%5E2+%2B+%28b-1%29%5E2+=+17%2F16 ALIGN=MIDDLE  ALT=\"%28a-%281%2F4%29%29%5E2+%2B+%28b-1%29%5E2+=+17%2F16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The center of the circle is at ((1/4),1); the radius is sqrt(17)/4.<BR>\n" );
document.write( " </SPAN>\n" );
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