document.write( "Question 1184217: If \"+%28z-1%29%2F%28z%2B1%29+\" is purely imaginary then what is value of |z|? \n" ); document.write( "
Algebra.Com's Answer #814755 by ikleyn(52803)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "            This problem has a nice geometric interpretation.\r
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document.write( "1 and -1 are selected points in the complex number plane.\r\n" );
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document.write( "For an arbitrary complex number z, the numbers z-1 and z+1 are vectors, connecting z with the points 1 and -1.\r\n" );
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document.write( "The condition that  \"%28z-1%29%2F%28z%2B1%29\"  is purely imaginary means that the vectors z-1 and z+1 are perpendicular.\r\n" );
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document.write( "So, the problem asks to find points z in complex plane, such that the \"visibility angle\" from z to \r\n" );
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document.write( "these points \"1\" and \"-1\" is the right angle.\r\n" );
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document.write( "Or, in other terms, find points z in complex plane, such that the angle between the vectors z-1 and z+1 is the right angle.\r\n" );
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document.write( "Clearly, these points are on the unit circle, and they provide the right angle leaning on the segment [-1,1]\r\n" );
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document.write( "as on the diameter of the unit circle.   So, |z| = 1.    ANSWER\r\n" );
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