document.write( "Question 625240: If the ellipse defined by the equation 16x^2+4y^2+96x-8y+84=0 is translated 6 units down and 7 units to the left, write the standard equation of the resulting ellipse \n" ); document.write( "

Algebra.Com's Answer #393398 by Edwin McCravy(20060) $\"\"$ $\"About$

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document.write( "16x² + 4y² + 96x - 8y + 84 = 0\r\n" );
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document.write( "Get the constant term off the left side by adding -84 to both sides\r\n" );
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document.write( "   16x² + 96x + 4y² - 8y = -84\r\n" );
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document.write( "Swap the two middle terms\r\n" );
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document.write( "   16x² + 96x + 4y² - 8y = -84\r\n" );
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document.write( "Factor out the coefficients of x² and y² \r\n" );
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document.write( "16(x² + 6x) + 4(y² - 2y) = -84\r\n" );
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document.write( "To complete the square inside the first parenhtheses,\r\n" );
document.write( "1. Multiply the coefficient of x, which is +6 by 1/2, getting 3\r\n" );
document.write( "2. Square this result, (3)² = +9\r\n" );
document.write( "3. Add +9 inside the first parentheses\r\n" );
document.write( "4. Multiply +9 by the coefficient we factored out, 16, getting +144\r\n" );
document.write( "5. Add +144 to the right side  \r\n" );
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document.write( "16(x² + 6x + 9) + 4(y² - 2y) = -84 + 144\r\n" );
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document.write( "To complete the square inside the second parenhtheses,\r\n" );
document.write( "1. Multiply the coefficient of y, which is -2 by 1/2, getting -1\r\n" );
document.write( "2. Square this result, (-1)² = +1\r\n" );
document.write( "3. Add +9 inside the first parentheses\r\n" );
document.write( "4. Multiply +1 by the coefficient we factored out, 4, getting +4\r\n" );
document.write( "5. Add +4 to the right side  \r\n" );
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document.write( "16(x² + 6x + 9) + 4(y² - 2y + 1) = -84 + 144 + 4\r\n" );
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document.write( "Factor the 1st parentheses:  x² + 6x + 9 = (x + 3)(x + 3) = (x + 3)²\r\n" );
document.write( "Factor the 2nd parentheses:  y² - 2x + 1 = (y - 1)(y - 1) = (y - 1)²\r\n" );
document.write( "Combine terms on the right:  -84 + 144 + 4 = 64\r\n" );
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document.write( "16(x + 3)² + 4(y - 1)² = 64\r\n" );
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document.write( "Get a 1 on the right side by dividing through by 64\r\n" );
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document.write( " +  = \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( "To translate this equation 6 units down and 7 units to the left,\r\n" );
document.write( "replace y by (y+6) and x by (x+7)\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( " +  = 1\r\n" );
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document.write( "Edwin

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