document.write( "Question 1152644: Find the value/s of U so that the graph of the equation of Ux^2 + y^2 − 2Ux = 0 is a hyperbola.\r
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Algebra.Com's Answer #774713 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
Find the value/s of U so that the graph of the equation of
\n" ); document.write( "Ux^2 + y^2 − 2Ux = 0 is a hyperbola.\r
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document.write( "That will be the equation of a hyperbola if you pick any negative number\r\n" );
document.write( "for U.  The easiest negative number to pick for U is -1:\r\n" );
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document.write( "\"Ux%5E2+%2B+y%5E2+%E2%88%92+2Ux+=+0\"\r\n" );
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document.write( "\"%28-1%29x%5E2+%2B+y%5E2+%E2%88%92+2%28-1%29x+=+0\"\r\n" );
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document.write( "\"-x%5E2+%2B+y%5E2+%2B+2x+=+0\"\r\n" );
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document.write( "\"y%5E2+-+x%5E2+%2B+2x+=+0\"\r\n" );
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document.write( "Write y² as (y-0)² and factor out a negative from the two terms in x:\r\n" );
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document.write( "\"%28y-0%29%5E2+-+%28x%5E2-2x%29=0\"\r\n" );
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document.write( "Complete the square by adding +1 inside the parentheses, which\r\n" );
document.write( "amounts to adding -1 to the left side, so add -1 to the right\r\n" );
document.write( "side also:\r\n" );
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document.write( "\"%28y-0%29%5E2+-+%28x%5E2-2x%2B1%29=-1\"\r\n" );
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document.write( "We write the second parentheses as the square of a binomial:\r\n" );
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document.write( "\"%28y-0%29%5E2+-+%28x-1%29%5E2=-1\"\r\n" );
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document.write( "Since the right side is negative, we multiply through by -1\r\n" );
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document.write( "\"-%28y-0%29%5E2+%2B+%28x-1%29%5E2=%2B1\"\r\n" );
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document.write( "We reverse the terms on the left:\r\n" );
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document.write( "\"%28x-1%29%5E2+-+%28y-0%29%5E2=1\"\r\n" );
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document.write( "We put 1's under the two terms on the left:\r\n" );
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document.write( "\"%28x-1%29%5E2%2F1%5E%22%22+-+%28y-0%29%5E2%2F1%5E%22%22=1\"\r\n" );
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document.write( "We compare \r\n" );
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document.write( "\"%28x-1%29%5E2%2F1%5E%22%22+-+%28y-0%29%5E2%2F1%5E%22%22=1\" to\r\n" );
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document.write( "\"%28x-h%29%5E2%2Fa%5E2+-+%28y-k%29%5E2%2Fb%5E2=1\"\r\n" );
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document.write( "The center is (h,k) = (1,0), a=1, b=1,\r\n" );
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document.write( "We calculate c, the distance fro center to foci from \r\n" );
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document.write( "\"c%5E2=a%5E2%2Bb%5E2\"\r\n" );
document.write( "\"c%5E2=1%5E2%2B1%5E2\"\r\n" );
document.write( "\"c%5E2=1%2B1\"\r\n" );
document.write( "\"c%5E2=2\"\r\n" );
document.write( "\"c=sqrt%282%29\"\r\n" );
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document.write( "So the foci are \"%28matrix%281%2C3%2C1%2Bsqrt%282%29%2C%22%2C%22%2C0%29%29\" and \"%28matrix%281%2C3%2C1-sqrt%282%29%2C%22%2C%22%2C0%29%29\",\r\n" );
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document.write( "so the distance between the foci is\r\n" );
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document.write( "\"2sqrt%282%29\"\r\n" );
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document.write( "If you chose any other negative number for U the answer would be different.\r\n" );
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document.write( "Edwin
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