Algebra.Com's Answer #83987 by Edwin McCravy(20056)![\"\"](\"/images/stars/stars-13.gif\")  
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I need to find the center,vertices, and foci of the ellipse
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document.write( "You first have to put it in standard form, which is either\r\n" );
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document.write( " (x - h)² (y - k)² \r\n" );
document.write( " ---------- + ---------- = 1 \r\n" );
document.write( " a² b²\r\n" );
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document.write( "if it turns out that the larger number is under\r\n" );
document.write( "the term on the left which contains x, and the \r\n" );
document.write( "graph will be an ellipse with a horizontal major\r\n" );
document.write( "axis and a vertical minor axis, where\r\n" );
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document.write( "1. The center is (h, k) \r\n" );
document.write( "2. The length of the semi-major axis is \" a \".\r\n" );
document.write( "3. The ends of the major axis are the points (h-a,k) and (h+a,k)\r\n" );
document.write( "4. The ends of the minor axis are the points (h,k-b) and (h,k+b)\r\n" );
document.write( "5. Foci are (h-c,k), (h+c,k) where c is calculated from c² = a²-b² \r\n" );
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document.write( "or\r\n" );
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document.write( " (x - h)² (y - k)² \r\n" );
document.write( " ---------- + ---------- = 1 \r\n" );
document.write( " b² a²\r\n" );
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document.write( "if it turns out that the larger number is under\r\n" );
document.write( "the term on the left which contains y, and the \r\n" );
document.write( "graph will be an ellipse with a horizontal major\r\n" );
document.write( "axis and a vertical minor axis, where and the \r\n" );
document.write( "graph will be an ellipse with a vertical major \r\n" );
document.write( "axis and a horizontal minor axis\r\n" );
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document.write( "where\r\n" );
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document.write( "1. The center is (h, k) \r\n" );
document.write( "2. The length of the semi-major axis is \" a \".\r\n" );
document.write( "3. The ends of the major axis are the points (h,k-a) and (h,k+a)\r\n" );
document.write( "4. The ends of the minor axis are the points (h-b,k) and (h+b,k)\r\n" );
document.write( "5. Foci are (h,k-a), (h,k+a) where c is calculated from c² = a²-b² \r\n" );
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document.write( "We can't tell which it is unti we get it into standard form:\r\n" );
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document.write( "4x² + 25y² - 8x + 100y + 4 = 0\r\n" );
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document.write( "Rearrange getting x² term then the x term then y²and y terms\r\n" );
document.write( "That is, swap the second and third terms:\r\n" );
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document.write( "4x² - 8x + 25y² + 100y + 4 = 0\r\n" );
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document.write( "Subtract 4 from both sides:\r\n" );
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document.write( " 4x² - 8x + 25y² + 100y = -4\r\n" );
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document.write( "Factor just the 4 out of the first two terms\r\n" );
document.write( "on the left and also factor just the 25 out of \r\n" );
document.write( "the last two terms on the left: \r\n" );
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document.write( " 4(x² - 2x) + 25(y² + 4y) = -4\r\n" );
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document.write( "Complete the square in the first parentheses:\r\n" );
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document.write( "The coefficient of x is -2\r\n" );
document.write( "Multiply it by  , getting -1\r\n" );
document.write( "Square -1, getting +1.\r\n" );
document.write( "Add +1 inside the first parentheses.\r\n" );
document.write( "But the parentheses has a 4 in front.\r\n" );
document.write( "So this amounts to adding 4·1 or 4 to\r\n" );
document.write( "the left side, so we must add +4 to\r\n" );
document.write( "the right side:\r\n" );
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document.write( " 4(x² - 2x + 1) + 25(y² + 4y) = -4 + 4\r\n" );
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document.write( "Complete the square in the second parentheses:\r\n" );
document.write( "The coefficient of y is 4\r\n" );
document.write( "Multiply it by , getting 2\r\n" );
document.write( "Square -2, getting +4.\r\n" );
document.write( "Add +4 inside the first parentheses.\r\n" );
document.write( "But the parentheses has a 25 in front.\r\n" );
document.write( "So this amounts to adding 25·4 or 100 to\r\n" );
document.write( "the left side, so we must add +100 to\r\n" );
document.write( "the right side:\r\n" );
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document.write( " 4(x² - 2x + 1) + 25(y² + 4y + 4) = -4 + 4 + 100\r\n" );
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document.write( "Factor both parentheses on the left and combine the \r\n" );
document.write( "numbers on the right\r\n" );
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document.write( " 4(x - 1)(x - 1) + 25(y + 2)(y + 2) = 100\r\n" );
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document.write( " 4(x - 1)² + 25(y + 2)² = 100\r\n" );
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document.write( "To get a 1 on the right side, divide every term\r\n" );
document.write( "by 100\r\n" );
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document.write( " 4(x - 1)² 25(y + 2)² 100\r\n" );
document.write( " --------- + ---------- = -----\r\n" );
document.write( " 100 100 100\r\n" );
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document.write( "Cancel \r\n" );
document.write( " 1 1 1 \r\n" );
document.write( " 4(x - 1)² 25(y + 2)² 100\r\n" );
document.write( " --------- + ---------- = -----\r\n" );
document.write( " 100 100 100\r\n" );
document.write( " 25 4 1 \r\n" );
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document.write( " (x - 1)² (y + 2)² \r\n" );
document.write( " ---------- + ---------- = 1 \r\n" );
document.write( " 25 4 \r\n" );
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document.write( "Since 25 is greater than 4, and since a² is always the\r\n" );
document.write( "larger or these in an ellipse, then a² is under the\r\n" );
document.write( "term which contains x, we compare that to:\r\n" );
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document.write( " (x - h)² (y - k)² \r\n" );
document.write( " ---------- + ---------- = 1 \r\n" );
document.write( " a² b²\r\n" );
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document.write( "so it is an ellipse with a horizontal major axis and\r\n" );
document.write( "a vertical minor axis. Comparing letters in this with\r\n" );
document.write( "the numbers in the equation we got, we see that:\r\n" );
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document.write( "h = 1, k = -2, a² = 25, b² = 4 so a = 5 and b = 2\r\n" );
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document.write( "This means\r\n" );
document.write( "1. The center is (h, k) = (1, -2)\r\n" );
document.write( "2. The length of the semi-major axis is a = 5\r\n" );
document.write( "3. The ends of the major axis (vertices) are the points (h-a,k), (h+a,k)\r\n" );
document.write( " or (1-5,-2) and (1+5,-2) or (-4,-2) and (6,-2)\r\n" );
document.write( "4. The ends of the minor axis are the points (h,k-b), (h,k+b)\r\n" );
document.write( " or (1,-2-2), (1,-2+2) or (1,-4) and (1,0)\r\n" );
document.write( "5. Foci are (h-c,k), (h+c,k) where c is gotten from c² = a²-b² \r\n" );
document.write( " We calculate c:\r\n" );
document.write( " c² = a² - b²\r\n" );
document.write( " c² = 5² - 2²\r\n" );
document.write( " c² = 25 - 4\r\n" );
document.write( " c² = 21\r\n" );
document.write( " c = Ö21 __ __ \r\n" );
document.write( " So foci are (h-c,k) and (h+c,k), or (1-Ö21,-2) and (1+Ö21), \r\n" );
document.write( " or about (-3.6,-2) and (5.6,-2)\r\n" );
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document.write( "We draw the major and minor axes and plot the foci: \r\n" );
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document.write( "Then we sketch in the ellipse:\r\n" );
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document.write( "The answers to your problem are:\r\n" );
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document.write( "center = (1, -2)\r\n" );
document.write( "vertices = ends of major axis = (-4,-2) and (6,-2)\r\n" );
document.write( "foci = (1-Ö21,-2) and (1+Ö21,-2), \r\n" );
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document.write( "Edwin \n" );
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