document.write( "Question 658496: a hyperbola of eccentricity 3/2 has one focus at (1,-3). The corresponding directrix is the line \"y\". find an equation for the hyperbola? \n" ); document.write( "

Algebra.Com's Answer #410318 by Edwin McCravy(20059) $\"\"$ $\"About$

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document.write( "You gave the corresponding directrix as just \"y\".  \r\n" );
document.write( "You have to give an equation, not just \"y\".  I will\r\n" );
document.write( "guess arbitrarily that you meant \"y=2\".  If you meant\r\n" );
document.write( "another number, then the principle will be the same.\r\n" );
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document.write( "                  distance from any pt. (x,y) to focus)  \r\n" );
document.write( "eccentricity = -------------------------------------------\r\n" );
document.write( "                distance from that pt. (x,y) to directrix)\r\n" );
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document.write( "Distance from (x,y) to focus (1,-3) = \r\n" );
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document.write( "Distance from (x,y) to the point (x,2) on the directrix = \r\n" );
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document.write( " =  = |y-2|\r\n" );
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document.write( "Use the eccentricity formula above:\r\n" );
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document.write( "                     = \r\n" );
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document.write( "Square both sides:\r\n" );
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document.write( "                     = \r\n" );
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document.write( "Cross-multiply:\r\n" );
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document.write( "            9(y - 2)² = 4[(x - 1)² + (y + 3)²]\r\n" );
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document.write( "       9(y² - 4y + 4) = 4(x - 1)² + 4(y + 3)²\r\n" );
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document.write( "       9y² - 36y + 36 = 4(x - 1)² + 4(y² + 6y + 9)\r\n" );
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document.write( "       9y² - 36y + 36 = 4(x - 1)² + 4y² +24y + 36\r\n" );
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document.write( "5y² - 60y - 4(x - 1)² = 0\r\n" );
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document.write( "Factor out 5 from the first two terms:\r\n" );
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document.write( "5(y² - 60y) - 4(x - 1)² = 0\r\n" );
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document.write( "Complete the square in the first parentheses:\r\n" );
document.write( "-60× = -30, the (-30)² = 900. Add 900 in the parentheses\r\n" );
document.write( "and since the parentheses has coefficient 5 we add 5·900 or 4500 \r\n" );
document.write( "on the right:\r\n" );
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document.write( "5(y² - 60y + 900) - 4(x - 1)² = 0 + 4500\r\n" );
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document.write( "5(y - 30)² - 4(x - 1)² = 4500\r\n" );
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document.write( "Get a 1 on the right by dividing through by 4500:\r\n" );
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document.write( " -  = 1\r\n" );
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document.write( " -  = 1\r\n" );
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document.write( "Edwin

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