document.write( "Question 826517: How to graph the ellipse y^2-6y+48x^2-3=0 \r
\n" ); document.write( "\n" ); document.write( "Help me please :(( \n" ); document.write( "
Algebra.Com's Answer #498120 by Edwin McCravy(20060) $\"\"$ $\"About$
You can put this solution on YOUR website!
\r\n" );
document.write( "y²-6y+48x²-3 = 0\r\n" );
document.write( "\r\n" );
document.write( "You have to do algebraic operations on it to get it in the form:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "in which case it will look like this ,\r\n" );
document.write( "\r\n" );
document.write( "or\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "in which case it will look like this \r\n" );
document.write( "\r\n" );
document.write( "We won't know which it is until later, when we can see which\r\n" );
document.write( "denominator is larger, for the larger denominator will be \"a²\" and \r\n" );
document.write( "the smaller denominator will be \"b²\".\r\n" );
document.write( "\r\n" );
document.write( "y²-6y+48x²-3 = 0\r\n" );
document.write( "\r\n" );
document.write( "Put the x term first, then the y terms second on the left \r\n" );
document.write( "of the equation and isolate the number on the right:\r\n" );
document.write( "\r\n" );
document.write( "48x²+y²-6y = 3\r\n" );
document.write( "\r\n" );
document.write( "Since there is no x-term, all we have to do is write the\r\n" );
document.write( "term 48x² as 48(x-0)².\r\n" );
document.write( "\r\n" );
document.write( "48(x-0)²+y²-6y = 3\r\n" );
document.write( "\r\n" );
document.write( "Since there is a y-term we must complete the square on\r\n" );
document.write( "y²-6y:\r\n" );
document.write( "\r\n" );
document.write( "To the side, we\r\n" );
document.write( "1. Multiply the coefficient of y, which is -6, by ,\r\n" );
document.write( "   getting -3.\r\n" );
document.write( "2. Square the result of step 1 (-3)² = 9\r\n" );
document.write( "3. Add +9 to both sides of the equation\r\n" );
document.write( "\r\n" );
document.write( "48(x-0)²+y²-6y+9 = 3+9\r\n" );
document.write( "\r\n" );
document.write( "Then we factor y²-6y+9 as (y-3)(y-3) and then as (y-3)²\r\n" );
document.write( "and replace y²-6y+9 by (y-3)².  Combine the numbers on\r\n" );
document.write( "the right side:\r\n" );
document.write( "\r\n" );
document.write( "48(x-0)²+(y-3)² = 12\r\n" );
document.write( "\r\n" );
document.write( "Then we divide through by 12 to get 1 on the right:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We must get the 48 off the top of the first fraction.\r\n" );
document.write( "To do that we divide top and bottom by 48:\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "Simplify:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now we can tell that it is an ellipse that looks like this \r\n" );
document.write( "\r\n" );
document.write( "because the larger denominator is a² which equals 12, and the\r\n" );
document.write( "smaller denominator is b² which equals \r\n" );
document.write( "\r\n" );
document.write( "So we compare it to:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(h,k) = (0,3) is the center, \r\n" );
document.write( "\r\n" );
document.write( "Since a² = 12 then a = √12 = √4*3 = 2√ ≈ 3.46\r\n" );
document.write( "\r\n" );
document.write( "Since b² =  then b =  = \r\n" );
document.write( "\r\n" );
document.write( "a = the distance from the center (0,3) to the vertices, and b = the distance\r\n" );
document.write( "from the center to the covertices.  So we plot the center, the vertices,\r\n" );
document.write( "which are a ≈ 3.46, units above and bellow the center, and the covertices\r\n" );
document.write( "which are  of a unit right and left of the center:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Then sketch in the ellipse (it's a tall skinny one!)\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );