document.write( "Question 1026729: Prove the proposition |z|=1 if {(1+z)/(1-z)} is purely imaginary. Please help. \n" ); document.write( "

Algebra.Com's Answer #642042 by richard1234(7193) $\"\"$ $\"About$

You can put this solution on YOUR website!
Using robertb's solution above:\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "We have

and

. Instead of solving for a and b explicitly and checking that |z| = 1, all we need to do is show that the system implies

.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Rearranging gives

and

. Squaring both equations and adding them gives\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

(note

terms cancel out)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\n" ); document.write( "

\n" );