document.write( "Question 1208498: if z=x+iy , z* conj(z)^3+conj(z)*z^3=350, then (x,y)=..[(2,1) ,(3,4) ,(4,3) ,(5,4)] \n" ); document.write( "

Algebra.Com's Answer #846951 by ikleyn(52778) $\"\"$ $\"About$

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\n" ); document.write( "if z=x+iy , z* conj(z)^3+conj(z)*z^3=350, then (x,y)=..[(2,1) ,(3,4) ,(4,3) ,(5,4)]
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document.write( "z*conj(z)^3+conj(z)*z^3 = z*conj(z)*(z^2 + (conj((z))^2) = |z|^2*(2x^2 - 2y^2).\r\n" );
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document.write( "So, our equation takes the form\r\n" );
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document.write( "      |z|^2*(2x^2 - 2y^2) = 350,  \r\n" );
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document.write( "or\r\n" );
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document.write( "     |z|^2*(x^2 - y^2) = 175.    (2)\r\n" );
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document.write( "In this problem,  |z|^2  and (x^2-y^2) are integer numbers;  so, our equation (2)  is an equation in integer numbers.\r\n" );
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document.write( "We have |z|^2 =  5  for  z = (2,1);\r\n" );
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document.write( "        |z|^2 = 25  for  z = (3,4);\r\n" );
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document.write( "        |z|^2 = 25  for  z = (4,3);\r\n" );
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document.write( "        |z|^2 = 41  for  z = (5,4).\r\n" );
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document.write( "Therefore, based on properties of divisibility, we see that (5,4) does not work.\r\n" );
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document.write( "Also,  x^2 - y^2 = 9 - 16  is negative number, so,  (3,4) does not work, too.\r\n" );
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document.write( "Next, for (2,1)  x^2 - y^2 = 2^2-1^2 = 4 - 1 = 3 does not divide 175, so (2,1) does not work.\r\n" );
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document.write( "The only remaining candidate is (4,3).  For it, |z|^2 = 25 and x^2 - y^2 = 16-9 = 7;  25*7 = 175.\r\n" );
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document.write( "So, the only pair which works is (4,3).\r\n" );
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document.write( "ANSWER.  Of four given pairs, the only solution to this problem is the pair (4,3).\r\n" );
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\n" ); document.write( "\n" ); document.write( "If to allow x and y in this problem to be arbitrary real numbers,
\n" ); document.write( "then the given equation would have infinitely many solutions.\r
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