document.write( "Question 52446: find the center, vertices, and foci of the ellipse 4x2+25y2-8x+100+4=0. graph \n" ); document.write( "
Algebra.Com's Answer #34986 by AnlytcPhil(1806)\"\" \"About 
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find the center, vertices, and foci of the ellipse\r\n" );
document.write( "4x² + 25y² - 8x + 100y + 4 = 0. graph\r\n" );
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document.write( "We have to get it in one of these two forms:\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( "which is an ellipse that looks like an egg\r\n" );
document.write( "sitting on a table, with the more pointed\r\n" );
document.write( "ends horizontal.\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( "which is an ellipse that looks like an egg\r\n" );
document.write( "which is being held with the more pointed ends\r\n" );
document.write( "vertical, that is, it looks like the character\r\n" );
document.write( "zero, \"0\". \r\n" );
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document.write( "We can tell this by looking to see whether the \r\n" );
document.write( "larger denominator is under the term with x and \r\n" );
document.write( "the smaller denominator is under the term in y, \r\n" );
document.write( "or vice-versa.  a is always larger than b, so\r\n" );
document.write( "a² is always larger than b², except when they\r\n" );
document.write( "are equal, (and then the ellipse is really a \r\n" );
document.write( "circle.  But they will not be equal here.)\r\n" );
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document.write( "4x² + 25y² - 8x + 100y + 4 = 0\r\n" );
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document.write( "Get the x terms and the y terms together in descending\r\n" );
document.write( "order:\r\n" );
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document.write( "4x² - 8x + 25y² + 100y + 4 = 0\r\n" );
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document.write( "Subtract the constant term 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 4 out of the first two terms on the left and \r\n" );
document.write( "factor out 25 out of 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( "Now inside each of those parenthese we want to add\r\n" );
document.write( "what is necessary to complete the square.\r\n" );
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document.write( "To complete the square inside (x² - 2x) we must\r\n" );
document.write( "multiply the coefficient of x by 1/2, then square it.\r\n" );
document.write( "So we multiply 2 by 1/2 and get 1, then when we\r\n" );
document.write( "square 1 we get 1.  So 1 is what we want to add\r\n" );
document.write( "inside the first parentheses.\r\n" );
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document.write( "To complete the square inside (y² + 4y) we must\r\n" );
document.write( "multiply the coefficient of y by 1/2, then square it.\r\n" );
document.write( "So we multiply 4 by 1/2 and get 2, then when we\r\n" );
document.write( "square 2 we get 4.  So 4 is what we want to add\r\n" );
document.write( "inside the second parentheses.\r\n" );
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document.write( "But when we add those numbers inside the parentheses\r\n" );
document.write( "we have no choice but to add whatever is necessary to\r\n" );
document.write( "to the right side also to offset it, so let's\r\n" );
document.write( "temporarily put blanks on the right and then we will\r\n" );
document.write( "see what we need to add to the right side to offset\r\n" );
document.write( "what we have added inside the two sets of parentheses:\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( "When we add 1 inside the first parentheses we really\r\n" );
document.write( "are adding 4×1 or 4, because the 4 on the outside of the\r\n" );
document.write( "first parentheses will multiply by the 1, showing that\r\n" );
document.write( "we really have added 4, not just 1, to the left side. So\r\n" );
document.write( "to offset we place 4 in the first blank on the right:\r\n" );
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document.write( "   4(x² - 2x + 1) + 25(y² + 4y + 4) = -4 +  4  +      \r\n" );
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document.write( "Similarly, when we add 4 inside the second parentheses \r\n" );
document.write( "we really are adding 25×4 or 100, because the 25 on the \r\n" );
document.write( "outside of the second parentheses will multiply by the 4,\r\n" );
document.write( "showing that we really have added 100, not just 4, to the\r\n" );
document.write( "left side, so to offset we place 100 in the second blank \r\n" );
document.write( "on the right:\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( "Now we factor each of the parentheses as a perfect square,\r\n" );
document.write( "and we combine the numbers on the right side:\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( "Now we divide every term by 100 to get 1 on the right\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( "Divide numerator and denominator of the first fraction by 4\r\n" );
document.write( "Divide numerator and denominator of the second fraction by 25\r\n" );
document.write( "The fraction on the right is just 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( "Now we can determine that a² = 25 and b² = 4 since a² must\r\n" );
document.write( "be larger than b², so this ellipse is horizontal, like an egg \r\n" );
document.write( "lying on a table, and NOT vertical like the number zero. So\r\n" );
document.write( "we compare it 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( "and see that the center = (h,k) = (1, -2).\r\n" );
document.write( "Since a² = 25, then a = 5 and that is the semi-major axis.\r\n" );
document.write( "Since b² = 4, then b = 2 and that is the semi-minor axis.\r\n" );
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document.write( "So we plot the center (1, -2).  Then we find the two vertices\r\n" );
document.write( "by finding the points which are a=5 units right and left of\r\n" );
document.write( "the center.  The point 5 units right of the center (1,-2) is\r\n" );
document.write( "the point (6,-2) and the point 5 units left of (1,-2) is the\r\n" );
document.write( "point (-4,-2),\r\n" );
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document.write( "Now we find the top and bottom points of the ellipse, by\r\n" );
document.write( "finding the points which are b=2 units directly above and \r\n" );
document.write( "directly below the center.  The point 2 units above the \r\n" );
document.write( "center (1,-2) is the point (1,0) and the point 2 units below\r\n" );
document.write( "the center (1,-2) is the point (1,-4).  \r\n" );
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document.write( "So after plotting those five points, like this\r\n" );
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document.write( "  \r\n" );
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document.write( "we sketch in the ellipse like this:\r\n" );
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document.write( "Now to find the foci.  These are two point inside the ellipse which are\r\n" );
document.write( "\"c\" units left and right of the center.  We have to calculate c from the\r\n" );
document.write( "equation c² = a² - b²\r\n" );
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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( "     __\r\n" );
document.write( "c = Ö21\r\n" );
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document.write( "So we find two points this distance left and right of the center\r\n" );
document.write( "(1,-2).  They are the points: \r\n" );
document.write( "    __              __\r\n" );
document.write( "(1+Ö21, -2) and (1-Ö21, -2) or about (-3.6, -2) and (5.6, -2)\r\n" );
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document.write( "Edwin
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