document.write( "Question 682002: z is a complex number such that the ratio z-i/z-1 is purely imaginary. Prove that z lies on a circle whose centre is at a point 1/2 + 1/2 i and whose radius is 1/v2( 1 divided by square root of 2) \n" ); document.write( "
Algebra.Com's Answer #422999 by Edwin McCravy(20055)\"\" \"About 
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document.write( "Since the ratio \"%28z-i%29%2F%28z-1%29\" is purely imaginary,\r\n" );
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document.write( "\"%28z-i%29%2F%28z-1%29\" = ki\r\n" );
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document.write( "z - i = ki(z - 1)\r\n" );
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document.write( "Substitute z = x + yi\r\n" );
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document.write( "x + yi - i = ki(x + yi - 1)\r\n" );
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document.write( "x + yi - i = kxi + kyi² - ki\r\n" );
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document.write( "x + yi - i = kxi + ky(-1) - ki\r\n" );
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document.write( "x + yi - i = kxi - ky - ki\r\n" );
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document.write( "Equate real parts:\r\n" );
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document.write( "(1)     x = -ky\r\n" );
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document.write( "Equate imaginary parts:\r\n" );
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document.write( "yi - i = kxi - ki\r\n" );
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document.write( "y - 1 = kx - k \r\n" );
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document.write( "(2)    y = kx + 1 - k\r\n" );
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document.write( "Substitute in (1)\r\n" );
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document.write( "      x = -k(kx + 1 - k)\r\n" );
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document.write( "      x = -k²x - k + k²\r\n" );
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document.write( "k²x + x = k² - k\r\n" );
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document.write( "x(k² + 1) = k² - k\r\n" );
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document.write( "        x = \"%28k%5E2-k%29%2F%28k%5E2%2B1%29\"\r\n" );
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document.write( "Substitute in (1)\r\n" );
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document.write( "        \"%28k%5E2-k%29%2F%28k%5E2%2B1%29\" = -ky\r\n" );
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document.write( "        \"%28k%28k-1%29%29%2F%28k%5E2%2B1%29\" = -ky\r\n" );
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document.write( "Divide both sides by -k\r\n" );
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document.write( "        \"%28-%28k-1%29%29%2F%28k%5E2%2B1%29\" = y\r\n" );
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document.write( "        \"%28-k%2B1%29%2F%28k%5E2%2B1%29\" = y\r\n" );
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document.write( "We need to show that the point (x,y) = (\"%28k%5E2-k%29%2F%28k%5E2%2B1%29\", \"%28-k%2B1%29%2F%28k%5E2%2B1%29\")\r\n" );
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document.write( "lies on the circle mentioned.    \r\n" );
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document.write( "The point \"1%2F2\"+\"1%2F2\"i is the point (\"1%2F2%29\", \"1%2F2\")\r\n" );
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document.write( "The circle with center (\"1%2F2%29\", \"1%2F2\") and radius \"1%2Fsqrt%282%29\"\r\n" );
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document.write( "is \r\n" );
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document.write( "{x - \"1%2F2\")² + (y - \"1%2F2\")² = \"%281%2Fsqrt%282%29%29%5E2\"\r\n" );
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document.write( "x² - x + \"1%2F4\" + y² - y + \"1%2F4\" = \"1%2F2\"\r\n" );
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document.write( "x² - x + y² - y = 0\r\n" );
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document.write( "Substitute (x,y) = (\"%28k%5E2-k%29%2F%28k%5E2%2B1%29\", \"%28-k%2B1%29%2F%28k%5E2%2B1%29\")\r\n" );
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document.write( "\"%28%28k%5E2-k%29%2F%28k%5E2%2B1%29%29%5E2\" - \"%28k%5E2-k%29%2F%28k%5E2%2B1%29\" + \"%28%28-k%2B1%29%2F%28k%5E2%2B1%29%29%5E2\" - \"%28-k%2B1%29%2F%28k%5E2%2B1%29\" = 0\r\n" );
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document.write( "\"%28k%5E2-k%29%5E2%2F%28k%5E2%2B1%29%5E2\" - \"%28k%5E2-k%29%2F%28k%5E2%2B1%29\" + \"%28-k%2B1%29%5E2%2F%28k%5E2%2B1%29%5E2\" - \"%28-k%2B1%29%2F%28k%5E2%2B1%29\" = 0\r\n" );
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document.write( "Get the LCD by multiplying the 2nd and 4th terms by \"%28k%5E2%2B1%29%2F%28k%5E2%2B1%29\"\r\n" );
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document.write( "\"%28k%5E2-k%29%5E2%2F%28k%5E2%2B1%29%5E2\" - \"%28%28k%5E2-k%29%28k%5E2%2B1%29%29%2F%28%28k%5E2%2B1%29%28k%5E2%2B1%29%29\" + \"%28-k%2B1%29%5E2%2F%28k%5E2%2B1%29%5E2\" - \"%28%28-k%2B1%29%28k%5E2%2B1%29%29%2F%28%28k%5E2%2B1%29%28k%5E2%2B1%29%29\" = 0\r\n" );
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document.write( "\"%28k%5E4-2k%5E3%2Bk%5E2%29%2F%28x%5E2%2B1%29%5E2\" - \"%28k%5E4-k%5E3%2Bk%5E2-k%29%2F%28k%2A2%2B1%29%5E2\" + \"%28k%5E2-2k%2B1%29%2F%28k%5E2%2B1%29%5E2\" - \"%28-k%5E3%2Bk%5E2-k%2B1%29%2F%28k%5E2%2B1%29%5E2\" = 0\r\n" );
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document.write( " = 0\r\n" );
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document.write( "\"0%2F%28k%5E2%2B1%29%5E2\" = 0\r\n" );
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document.write( "0 = 0\r\n" );
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document.write( "Therefore the point (x,y) = (\"%28k%5E2-k%29%2F%28k%5E2%2B1%29\", \"%28-k%2B1%29%2F%28k%5E2%2B1%29\")\r\n" );
document.write( " lies on the circle with center (\"1%2F2%29\", \"1%2F2\") and radius \"1%2Fsqrt%282%29\" \r\n" );
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document.write( "Edwin
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