document.write( "Question 1041900: Determine the center and radii of the ellipse 2x^2+8x+3y^2-12y=10 \n" ); document.write( "
Algebra.Com's Answer #656828 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "We get it into one of the forms:\r\n" );
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document.write( "\"%28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2+=+1\" or \"%28x-h%29%5E2%2Fb%5E2%2B%28y-k%29%5E2%2Fa%5E2+=+1\"\r\n" );
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document.write( "2x²+8x+3y²-12y = 10\r\n" );
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document.write( "2(x²+4x)+3(y²-4y) = 10\r\n" );
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document.write( "Take half of 4, get 2.\r\n" );
document.write( "Square 2, get 4.\r\n" );
document.write( "Add 4 inside the first parentheses. \r\n" );
document.write( "Since the first parentheses is \r\n" );
document.write( "multiplied by 2, adding 2 inside\r\n" );
document.write( "the first parentheses amounts to\r\n" );
document.write( "adding 2 times 4, or 8 to the left\r\n" );
document.write( "side.  So add 8 to the right side\r\n" );
document.write( "to offset.\r\n" );
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document.write( "2(x²+4x+4)+3(y²-4y) = 10+8\r\n" );
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document.write( "Take half of -4, get -2.\r\n" );
document.write( "Square -2, get 4.\r\n" );
document.write( "Add 4 inside the second parentheses. \r\n" );
document.write( "Since the second parentheses is \r\n" );
document.write( "multiplied by 3, adding 4 inside\r\n" );
document.write( "the second parentheses amounts to\r\n" );
document.write( "adding 3 times 4, or 12 to the left\r\n" );
document.write( "side.  So add 12 to the right side\r\n" );
document.write( "to offset.\r\n" );
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document.write( "2(x²+4x+4)+3(y²-4y+4) = 10+8+12\r\n" );
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document.write( "Factor each parenthetical expression\r\n" );
document.write( "as the square of a binomial and collect\r\n" );
document.write( "terms on the right.\r\n" );
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document.write( "2(x+2)²+3(y-2)² = 30\r\n" );
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document.write( "Get 1 on the right side by dividing\r\n" );
document.write( "every term by 30\r\n" );
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document.write( "\"2%28x%2B2%29%5E2%2F30%2B3%28y-2%29%5E2%2F30+=+30%2F30\"\r\n" );
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document.write( "\"%28x%2B2%29%5E2%2F15%2B%28y-2%29%5E2%2F10+=+1\"\r\n" );
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document.write( "Since the larger denominator is under the \r\n" );
document.write( "term in x, we can tell that the ellipse \r\n" );
document.write( "looks like this \"drawing%2820%2C10%2C-2%2C2%2C-1%2C1%2Carc%280%2C0%2C-3.9%2C1.9%29+%29\"\r\n" );
document.write( " and not like this \"drawing%2810%2C20%2C-1%2C1%2C-2%2C2%2Carc%280%2C0%2C1.9%2C-3.9%29+%29\".\r\n" );
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document.write( "We compare it to the standard form for an ellipse:\r\n" );
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document.write( "\"%28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2+=+1\"\r\n" );
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document.write( "h=-2, k=2, a²=15, b²=10\r\n" );
document.write( "So the center is (h,k) = (-2,2) and\r\n" );
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document.write( "\"a=sqrt%2815%29\", \"b=sqrt%2810%29\"\r\n" );
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document.write( "Those are the semi-major and semi-minor axes. You\r\n" );
document.write( "call them long and short radii?\r\n" );
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document.write( "Edwin
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