document.write( "Question 738632: I need help finding a) vertex b) vertices c) foci d) major axis and e) minor axis of the ellipse: 9x^2-18x+4y^2-27=0 \n" ); document.write( "
Algebra.Com's Answer #450882 by Ed Parker(21) $\"\"$ $\"About$
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document.write( "The idea is to get the equation like one of these\r\n" );
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document.write( "a is one-half the major axis and b is one-half the minor axis.\r\n" );
document.write( "So a² is always larger than b².  If a² is under the term in x,\r\n" );
document.write( "the ellipse is like this:  and if b² is under the term in x, \r\n" );
document.write( "the ellipse looks like this .\r\n" );
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document.write( " 9x² - 18x + 4y² - 27 = 0\r\n" );
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document.write( "Add 28 to both sides to get the constant \r\n" );
document.write( "term off the left side:\r\n" );
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document.write( "      9x² - 18x + 4y² = 27\r\n" );
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document.write( "Factor 9 out of of the first two terms:\r\n" );
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document.write( "     9(x² - 2x) + 4y² = 27\r\n" );
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document.write( "Complete the square in the parentheses:\r\n" );
document.write( "1. Multiply the coefficient of x, which is -2 \r\n" );
document.write( "   by , getting -1\r\n" );
document.write( "2. Square -1, getting (-1)² or +1\r\n" );
document.write( "3. Add +1 to the expression in the parentheses.\r\n" );
document.write( "4. Since the parentheses is multiplied by 9,\r\n" );
document.write( "   this amouts to adding 9 to the left side,\r\n" );
document.write( "   so we must add 9·1 or 9 to the right side.\r\n" );
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document.write( " 9(x² - 2x + 1) + 4y² = 27 + 9\r\n" );
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document.write( "We factor x² - 2x + 1 as (x - 1)(x - 1) and as (x - 1)²\r\n" );
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document.write( "      9(x - 1)² + 4y² = 36\r\n" );
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document.write( "There is no y term so we do not need to complete\r\n" );
document.write( "the square.  We just write y as (y - 0)\r\n" );
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document.write( "9(x - 1)² + 4(y - 0)² = 36  \r\n" );
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document.write( "Next we get 1 on the right side by dividing every \r\n" );
document.write( "term by 36\r\n" );
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document.write( "Simplify:\r\n" );
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document.write( "Since 4 is less than 9 we know that b²=4 is under the \r\n" );
document.write( "term in x, so the ellipse looks like this .  We compare that to\r\n" );
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document.write( "and get h=1, b²=4, k=0, a²=9, and of course b=2 and a=3\r\n" );
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document.write( "So the center = (h,k) = (1,0), the major axis is vertical, a=3 units\r\n" );
document.write( "above the center and a=3 units below the center, like this green line,\r\n" );
document.write( "which is the whole major axis:\r\n" );
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document.write( "The minor axis is horizontal, b=2 units left of the center and b=2\r\n" );
document.write( "units right of the center, like this green line, which is the whole\r\n" );
document.write( "minor axis:\r\n" );
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document.write( "So we can sketch in the ellipse:\r\n" );
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document.write( " To calculate the foci, we need the quantity c, the distance\r\n" );
document.write( "from the center to the foci, given by the equation:\r\n" );
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document.write( "c² = a² - b²\r\n" );
document.write( "c² = 9 - 4\r\n" );
document.write( "c² = 5\r\n" );
document.write( " c = √5\r\n" );
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document.write( "So the foci are the points (1,±√5)\r\n" );
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document.write( "The foci are plotted below:\r\n" );
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document.write( "The center is (1,0). The vertices are (1,±3).\r\n" );
document.write( "the covertices are (-1,0) and (3,0),\r\n" );
document.write( "the major axis is 6, the minor axis is 4.\r\n" );
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document.write( "Edwin
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