document.write( "Question 1060958: what is the length of the major axis of the ellipse X2+9Y2-2X+18Y+1=0? \n" ); document.write( "

Algebra.Com's Answer #675798 by KMST(5328) $\"\"$ $\"About$

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$\"x%5E2%2B9y%5E2-2x%2B18y%2B1=0\"$
\n" ); document.write( " $\"x%5E2-2x%2B9y%5E2%2B18y=-1\"$
\n" ); document.write( " $\"x%5E2-2x%2B1%2B9y%5E2%2B18y%2B9=-1%2B1%2B9\"$
\n" ); document.write( " $\"%28x-1%29%5E2%2B9%28y%5E2%2B2y%2B1%29=9\"$
\n" ); document.write( " $\"%28x-1%29%5E2%2B9%28y%2B1%29%5E2=9\"$
\n" ); document.write( "You can continue to
\n" ); document.write( " $\"%28x-1%29%5E2%2F9%2B%28y%2B1%29%5E2=1\"$ ,
\n" ); document.write( "but even before that last step, it is obvious that
\n" ); document.write( " $\"%28x-1%29%5E2%3C=9\"$ while $\"%28y%2B1%29%5E2%3C=1\"$ ,
\n" ); document.write( "so $\"x-1\"$ ranges from $\"-3\"$ to $\"3\"$ ,
\n" ); document.write( "a difference of $\"highlight%286%29\"$ ,
\n" ); document.write( "and of course the extreme values of $\"x\"$ also have a difference of $\"6\"$ .
\n" ); document.write( "The length of the major axis is $\"highlight%286%29\"$ .
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\n" ); document.write( "NOTES:
\n" ); document.write( "The last two equations show that the ellipse is symmetrical with respect to $\"x=1\"$ and $\"y=-1\"$ , so (1,-1) is the center.
\n" ); document.write( "In general, an ellipse can be written in the form
\n" ); document.write( " $\"%28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2=1\"$ , or $\"%28x-h%29%5E2%2Fb%5E2%2B%28y-k%29%5E2%2Fa%5E2=1\"$ ,
\n" ); document.write( "and once you get to that form,
\n" ); document.write( "choosing $\"a\"$ and $\"b\"$ so that $\"a%3Eb\"$ ,
\n" ); document.write( " $\"a\"$ is the semi-major axis, which is parallel to the x- or y-axis (whatever variable is on to of the $\"a%5E2\"$ ),
\n" ); document.write( " $\"b\"$ is the semi-minor axis,
\n" ); document.write( " $\"%22%28+h+%2C+k+%29%22\"$ is the center,
\n" ); document.write( "and you can find the foci knowing that
\n" ); document.write( "they are on the major axis, at a distance $\"c\"$ to each side of the center, with $\"a%5E2=b%5E2%2Bc%5E2\"$ . \n" ); document.write( "

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