Algebra.Com's Answer #688242 by ikleyn(52788)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "Circle T intersects the hyperbola y=1/x at (1,1), (3, 1/3), and two other points.
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document.write( "What is the product of the y coordinates of the other two points? Please write in proof format. Thank you.
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document.write( "The equation of the circle is\r\n" );
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document.write( " =  ,\r\n" );
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document.write( "for some \"a\", \"b\" and \"r\".\r\n" );
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document.write( "The equation of the hyperbola is y =  (given !)\r\n" );
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document.write( "You will get the equation for common points (intersection points) if you substitute equation (2) into the equation (1).\r\n" );
document.write( "You will get\r\n" );
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document.write( " +  = , or\r\n" );
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document.write( " +  -  +  = 0.\r\n" );
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document.write( "Next multiply both sides by  to rid of denominators. You will get\r\n" );
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document.write( " = 0, or, ordering by descending degrees of x\r\n" );
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document.write( " = 0.\r\n" );
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document.write( "The last equation is the 4-th degree equation. Its roots are x-coordinates of the common (intersection) points.\r\n" );
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document.write( "Two of the roots are given: they are x-coordinates of the given intersection points x= 1 and x= 3.\r\n" );
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document.write( "Two other roots are not known.\r\n" );
document.write( "But, according to the Vieta's theorem for the equation of the degree 4, the product of four roots is the constant term \r\n" );
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document.write( " ( ! - it is the KEY idea ! ).\r\n" );
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document.write( "Thus,  =  = 1, (1)\r\n" );
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document.write( "which implies\r\n" );
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document.write( "  =  = 3. (2)\r\n" );
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document.write( "The problem asks about  , but it is simply \r\n" );
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document.write( " =  . =  =  \r\n" );
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document.write( "due to (2).\r\n" );
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document.write( "So, the problem is solved and the answer is: the product of y-coordinates of the two other intersection points is .\r\n" );
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document.write( "Answer. The product of y-coordinates of the two other intersection points is .\r\n" );
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document.write( "Solved.\r
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document.write( "For Vieta's Theorem see this Wikipedia article.\r
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