document.write( "Question 19417: The Equation $\"9x%5E2-18x%2B4y%5E2%2B8y=23\"$ defines an ellipse with center (____, ___). The major axis has length ___ and the minor axis has length ____. \n" ); document.write( "

Algebra.Com's Answer #9397 by Earlsdon(6294) $\"\"$ $\"About$

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You will need to get your equation into the standard form for an ellipse:
\n" ); document.write( " $\"%28%28x-h%29%5E2%29%2Fa+%2B+%28%28y-k%29%5E2%29%2Fb+=+1\"$
\n" ); document.write( "You can do this by completing the square in the x-terms and the y-terms. Start by groupng the x-terms together and grouping the y-terms together.\r
\n" ); document.write( "\n" ); document.write( " $\"%289x%5E2+-+18x%29+%2B+%284y%5E2+%2B+8y%29+=+23\"$ Now,factor a 9 from the x-group and factor a 4 from the y-group.
\n" ); document.write( " $\"9%28x%5E2+-+2x%29+%2B+4%28y%5E2+%2B+2y%29+=+23\"$ Complete the square in the x-terms within the parentheses by adding the square of half the x-coefficient to both sides of the equation, and do the same for the y-terms.
\n" ); document.write( " $\"9%28x%5E2+-+2x+%2B+1%29+%2B+4%28y%5E2+%2B+2y+%2B+1%29+=+23+%2B+9+%2B+4\"$
\n" ); document.write( " $\"9%28x%5E2+-+2x+%2B+1%29+%2B+4%28y%5E2+%2B+2y+%2B+1%29+=+36\"$ Now divide both sides by 36.
\n" ); document.write( " $\"%28x%5E2+-+2x+%2B+1%29%2F4+%2B+%28y%5E2+%2B+2y+%2B+1%29%2F9+=+1\"$ Factor the denominators.
\n" ); document.write( " $\"%28%28x+-+1%29%5E2%29%2F4+%2B+%28%28y+%2B+1%29%5E2%29%2F9+=+1\"$
\n" ); document.write( "This is the standard form of the equation for an ellipse with the center at (1, -1) and major axis is parallel to the y-axis. The length of the major axis is 3 and the length of the minor axis is 2 \n" ); document.write( "

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