document.write( "Question 1116382: (z-1)/(z+1) = ki
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Algebra.Com's Answer #731279 by ikleyn(52835)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "          If \"%28z-1%29%2F%28z%2B1%29\" = ki,   where k is a real number,  then show that |z| = 1.\r\n" );
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        to be correct   (adding that k is a real number).\r
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document.write( "Let  z = a + bi.\r\n" );
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document.write( "We are given  \"%28z-1%29%2F%28z%2B1%29\" = ki,  which means that\r\n" );
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document.write( "\"%28a%2Bbi-1%29%2F%28a%2Bbi%2B1%29\" = ki.\r\n" );
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document.write( "Left side is\r\n" );
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document.write( "\"%28a%2Bbi-1%29%2F%28a%2Bbi%2B1%29\" = \"%28a%2Bbi-1%29%2F%28a%2Bbi%2B1%29\".\"%28a-bi%2B1%29%2F%28a-bi%2B1%29\" = \"%28%28%28a-1%29%2Bbi%29%2A%28%28a%2B1%29-bi%29%29%2F%28%28a%2B1%29%5E2%2Bb%5E2%29\".\r\n" );
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document.write( "The denominator is now a real number.\r\n" );
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document.write( "The numerator is  (a-1)*(a+1) + bi*(a+1) - bi*(a-1) + b^2.\r\n" );
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document.write( "Since the ratio  \"Num%2FDen\" is purely imaginary number ki,  it means that the real part of the numerator is zero:\r\n" );
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document.write( "    (a-1)*(a+1) + b^2 = 0,   or\r\n" );
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document.write( "    a^2 - 1 + b^2 = 0,  which is equivalent to\r\n" );
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document.write( "    a^2 + b^2 = 1.\r\n" );
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document.write( "    The last equality precisely means that  |z| = a^2 + b^2 = 1,   QED.\r\n" );
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