document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=3449>Question 3449</A>: <I><SPAN STYLE=\"word-break: break-all;\"> z is an element of C such that [(z)/(z-i)] is real.How can I show that z is imaginary. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #1527 by </B><A HREF=/tutors/aboutme.mpl?userid=khwang><B>khwang(438)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=khwang><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=1527\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  Let z = a+bi where a,b are real.<BR>\n" );
document.write( " If z/(z-i) = c for some real c.<BR>\n" );
document.write( " then a + b i = c(z-i)<BR>\n" );
document.write( " or a + b i = c(a+(b-1)i)<BR>\n" );
document.write( " or a + b i = ca + c(b-1)i<BR>\n" );
document.write( " So, a=ca and b = c(b-1)\r<BR>\n" );
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document.write( " a=ca implies a(c-1) = 0 Hence, a=0 or c =1\r<BR>\n" );
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document.write( " But, if c = 1, then b = c(b-1) implies b = b-1 impossible.\r<BR>\n" );
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document.write( " Hence, we see that a = 0, z = bi <BR>\n" );
document.write( " If b = 0, then z = 0 and c = 0<BR>\n" );
document.write( "  and so z is imaginary or zero.\r<BR>\n" );
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document.write( " Kenny </SPAN>\n" );
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