document.write( "Question 299505: find the vertices, center, and foci of the ellipse, and sketc its graph 9x^2+4y^2-36x+8y+31=0 \n" ); document.write( "
Algebra.Com's Answer #215198 by Edwin McCravy(20060)\"\" \"About 
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find the vertices, center, and foci of the ellipse, and sketc its graph 9x^2+4y^2-36x+8y+31=0
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document.write( "Rearrange\r\n" );
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document.write( "\"9x%5E2-36x%2B4y%5E2%2B8y=-31\"\r\n" );
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document.write( "Factor out coefficients of squared letters:\r\n" );
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document.write( "\"9%28x%5E2-4x%29%2B4%28y%5E2%2B2y%29=-31\"\r\n" );
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document.write( "Complete the square in the first parentheses by\r\n" );
document.write( "adding \"red%28%22%2B4%22%29\" inside the first parentheses\r\n" );
document.write( "which actually amounts to adding 36 to the left side \r\n" );
document.write( "because there is a 9 in front of the parentheses, so\r\n" );
document.write( "we must add a 36 to the right side:\r\n" );
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document.write( "\"9%28x%5E2-4x%2Bred%284%29%29%2B4%28y%5E2%2B2y%29=-31%2Bred%2836%29\"\r\n" );
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document.write( "Complete the square in the second parentheses by\r\n" );
document.write( "adding \"red%28%22%2B1%22%29\" inside the second parentheses\r\n" );
document.write( "which actually amounts to adding 4 to the left side \r\n" );
document.write( "because there is a 4 in front of the parentheses, so\r\n" );
document.write( "we must add a 4 to the right side:\r\n" );
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document.write( "\"9%28x%5E2-4x%2Bred%284%29%29%2B4%28y%5E2%2B2y%2Bred%281%29%29=-31%2Bred%2836%29%2Bred%284%29\"\r\n" );
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document.write( "Factoring the parentheses as a perfect squares:\r\n" );
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document.write( "\"9%28x-2%29%5E2%2B4%28y%2B1%29=9\"\r\n" );
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document.write( "Get a 1 on the right by dividing through by 9\r\n" );
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document.write( "\"%289%28x-2%29%5E2%29%2F9%2B%284%28y%2B1%29%5E2%29%2F9=9%2F9\"\r\n" );
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document.write( "\"%28x-2%29%5E2%2F1%2B4%28y%2B1%29%5E2%2F9=1\"\r\n" );
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document.write( "To get the 4 off the top of the second fraction we \r\n" );
document.write( "divide top and bottom by 4:\r\n" );
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document.write( "\"%28x-2%29%5E2%2F1%2B%284%28y%2B1%29%5E2%2F4%29%2F%289%2F4%29=1\"\r\n" );
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document.write( "\"%28x-2%29%5E2%2F1%2B%28cross%284%29%28y%2B1%29%5E2%2Fcross%284%29%29%2F%289%2F4%29=1\"\r\n" );
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document.write( "\"%28x-2%29%5E2%2F1%2B%28y%2B1%29%5E2%2F%289%2F4%29=1\"\r\n" );
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document.write( "Since the largest denominator is under the term in\r\n" );
document.write( "y, the ellipse has a vertical major axis.  So we\r\n" );
document.write( "compare it to:\r\n" );
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document.write( "\"%28x-h%29%5E2%2Fb%5E2%2B%28y-k%29%5E2%2Fa%5E2=1\"\r\n" );
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document.write( "\"h=2\", \"k=-1\", \r\n" );
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document.write( "\"b%5E2=1\" so \"b=sqrt%281%29=1\"\r\n" );
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document.write( "\"a%5E2=9%2F4\" so \"a=sqrt%289%2F4%29=3%2F2\"\r\n" );
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document.write( "Its center is at (h,k) = (1,\r\n" );
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document.write( "Draw the major axis \"a=3%2F2\" units both above and below the center.\r\n" );
document.write( "Draw the minor axis \"b=1\" units both right and left of the center.\r\n" );
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document.write( "The vertices are \"a=3%2F2\" units above and below the center (2,-1)\r\n" );
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document.write( "So we add \"3%2F2\" to the y-coordinate of the center\r\n" );
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document.write( "\"-1%2B3%2F2=-2%2F2%2B3%2F2=1%2F2\" so the upper vertex is (2,\"1%2F2\")\r\n" );
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document.write( "And we subtract \"3%2F2\" from the y-coordinate of the center\r\n" );
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document.write( "\"-1-3%2F2=-2%2F2-3%2F2=-5%2F2\" so the lower vertex is (2,\"-5%2F2\")\r\n" );
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document.write( "Sketch in the ellipse:\r\n" );
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document.write( "To find the foci, we must calculate c, using the Pythagorean\r\n" );
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document.write( "\"c%5E2=a%5E2-b%5E2\"\r\n" );
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document.write( "\"c%5E2=%283%2F2%29%5E2-1%5E2\"\r\n" );
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document.write( "\"c%5E2=9%2F4-1\"\r\n" );
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document.write( "\"c%5E2=9%2F4-4%2F4\"\r\n" );
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document.write( "\"c%5E2=5%2F4\"\r\n" );
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document.write( "\"c=sqrt%285%29%2F2\"\r\n" );
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document.write( "The foci are \"c=sqrt%285%29%2F2\" units above and below the center (2,-1)\r\n" );
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document.write( "So we add \"c=sqrt%285%29%2F2\" to the y-coordinate of the center\r\n" );
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document.write( "\"-1%2Bsqrt%285%29%2F2\" so the upper vertex is (2,\"-1%2Bsqrt%285%29%2F2\")\r\n" );
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document.write( "And we subtract \"c=sqrt%2813%29%2F2\" from the y-coordinate of the center\r\n" );
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document.write( "\"-1-sqrt%285%29%2F2\" so the upper vertex is (2,\"-1-sqrt%285%2F2%29\")\r\n" );
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document.write( "They are approximately \r\n" );
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document.write( "(1,0.12) and (1,-2.12)\r\n" );
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document.write( "We draw them in\r\n" );
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document.write( "Edwin
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