document.write( "Question 472808: I need to know what is the center, vertices, and foci of this ellipse
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Algebra.Com's Answer #324161 by Edwin McCravy(20056)\"\" \"About 
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document.write( "4x² + y² - 8x - 2y + 1 = 0\r\n" );
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document.write( "Get the constant off the left side:\r\n" );
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document.write( "    4x² + y² - 8x - 2y = -1\r\n" );
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document.write( "Swap the 2nd and 3rd terms\r\n" );
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document.write( "    4x² - 8x + y² - 2y = -1\r\n" );
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document.write( "Out of the first two terms factor out the coefficient of x²\r\n" );
document.write( "which is 4\r\n" );
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document.write( "  4(x² - 2x) + y² - 2y = -1\r\n" );
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document.write( "We complete the square inside the parentheses:\r\n" );
document.write( "Multiply the coefficient of x, which is -2, by \"1%2F2\",\r\n" );
document.write( "getting -1, Then square that result, (-1)² = 1. So we add\r\n" );
document.write( "1 inside the parentheses, getting 4(x² - 2x + 1), which,\r\n" );
document.write( "because of the 4*1 in front amount to adding 4 to the left\r\n" );
document.write( "side so we have to add 4 to the right side:\r\n" );
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document.write( "  4(x² - 2x + 1) + y² - 2y = -1 + 4\r\n" );
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document.write( "Next we complete the square for the y terms:\r\n" );
document.write( "Multiply the coefficient of y, which is -2, by \"1%2F2\",\r\n" );
document.write( "getting -1, Then square that result, (-1)² = 1. So we add\r\n" );
document.write( "1 to both sides:\r\n" );
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document.write( "  4(x² - 2x + 1) + y² - 2y + 1 = -1 + 4 + 1\r\n" );
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document.write( "Now we factor \"x²-2x+1\" as (x-1)² and \"y²-2y+1\" as \"(y-1)²\"\r\n" );
document.write( "and combine the numbers on the right as 4\r\n" );
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document.write( "  4(x - 1)² + (y - 1)² = 4\r\n" );
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document.write( "Then we divide through by 4 to get 1 on the right\r\n" );
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document.write( "  \"4%28x+-+1%29%B2%2F4+%2B+%28y+-+1%29%B2%2F4+=+4%2F4\"\r\n" );
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document.write( "  \"%28x-1%29%B2%2F1+%2B+%28y-1%29%2F4+=+1\"\r\n" );
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document.write( "Compare that to the standard form:\r\n" );
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document.write( "  \"%28x-h%29%B2%2Fb%B2+%2B+%28y-k%29%2Fa%B2+=+1\"\r\n" );
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document.write( "since a > b, so a² > b² in an ellipse. a=1, b=2\r\n" );
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document.write( "So this is an ellipse with center (h,k) = (1,1),\r\n" );
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document.write( "That's one question answered:  center = (1,1)\r\n" );
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document.write( "Major axis vertical which is 2a or 2(2) or 4\r\n" );
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document.write( "Minor axis horizontal which is 2b or 2(1) = 2\r\n" );
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document.write( "So we draw the center (1,1) and the major and minor axes,\r\n" );
document.write( "respective 4 and 2 units long with the center (1,1) as the\r\n" );
document.write( "midoint of each.\r\n" );
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document.write( "Then we draw in the ellipse:\r\n" );
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document.write( "The vertices are at the top and bottom.  They are (1,3) and (1,-1)\r\n" );
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document.write( "We have to calculate the focal points which are (h,k±c) inside \r\n" );
document.write( "the ellipse on the major axis:\r\n" );
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document.write( "First we calculate c from\r\n" );
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document.write( "c² = a² - b²\r\n" );
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document.write( "c² = 2² - 1²\r\n" );
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document.write( "c² = 4 - 1\r\n" );
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document.write( "c² = 3\r\n" );
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document.write( "c = ±Ö3\r\n" );
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document.write( "So the foci are (1,1+Ö3) and (1,1-Ö3). They are the two points plotted\r\n" );
document.write( "below inside the ellipse on the major axis.\r\n" );
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document.write( "Edwin
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