document.write( "Question 1183607: Locate the center, vertices, the foci, and the ends of the latera recta then graph the ellipse whose equation is 4x²+9y²-16x+18y-11=0. \n" ); document.write( "

Algebra.Com's Answer #814022 by KMST(5328) $\"\"$ $\"About$

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$\"4x%5E2%2B9y%5E2-16x%2B18y-11=0\"$
\n" ); document.write( " $\"4%28x%5E2-4x%29%2B9%28y%5E2%2B2y%29=11\"$
\n" ); document.write( " $\"4%28x%5E2-4x%2B4%29%2B9%28y%5E2%2B2y%2B1%29=11%2B4%2A4%2B9%2A1\"$
\n" ); document.write( " $\"4%28x-2%29%5E2%2B9%28y%2B1%29%5E2=11%2B16%2B9\"$
\n" ); document.write( " $\"4%28x-2%29%5E2%2B9%28y%2B1%29%5E2=36\"$
\n" ); document.write( "The equation above shows that that $\"system%28x-2=0%2Cy%2B1=0%29\"$ or $\"system%28x=2%2Cy=-1%29\"$ are axes (of symmetry) of the ellipse.
\n" ); document.write( "The center of the ellipse is the point where they intersect:
\n" ); document.write( "the point $\"P%282%2C-1%29\"$
\n" ); document.write( "Your teacher may like to write the equation as $\"%28x-2%29%5E2%2F9%2B%28y%2B1%29%5E2%2F4=1\"$ or $\"%28x-2%29%5E2%2F3%5E2%2B%28y%2B1%29%5E2%2F2%5E2=1\"$ ,
\n" ); document.write( "dividing by $\"36\"$ both sides of
\n" ); document.write( " $\"4%28x-2%29%5E2%2B9%28y%2B1%29%5E2=36\"$ .
\n" ); document.write( "They are just equivalent equations for the same ellipse.
\n" ); document.write( "You could say they show more clearly that for all the points of the ellipse
\n" ); document.write( " $\"%28x-2%29%5E2%2F9%3C=1\"$ --> $\"abs%28x-2%29%3C=3\"$ and $\"%28y%2B1%29%5E2%2F4%3C=1\"$ --> $\"abs%28y%2B1%29%3C=2\"$
\n" ); document.write( "They also show that the vertices of the ellipse are on the axes at $\"system%28y=-1%2Cx=2+%2B-+3%29\"$ and $\"system%28x=2%2Cy=-1+%2B-+2%29\"$
\n" ); document.write( "at $\"A%28-1%2C-1%29\"$ , $\"B%285%2C-1%29\"$ , $\"C%282%2C1%29\"$ , and $\"D%282%2C-3%29\"$ .
\n" ); document.write( "The segment AB, on the \"horizontal\" $\"y=-1\"$ axis, is called the major axis, because for this ellipse it is longer, going $\"a=3\"$ units to left and right of the center of the ellipse.
\n" ); document.write( "The segment CD, on the \"vertical\" $\"x=2\"$ axis, is called the minor axis, because for this ellipse it is shorter, going $\"b=2\"$ units up and down from the center of the ellipse.
\n" ); document.write( "The foci are on the major axis at a distance $\"c\"$ to both sides of the center, and we calculate $\"c\"$ from $\"a%5E2=b%5E2%2Bc%5E2\"$ .
\n" ); document.write( " $\"c=sqrt%283%5E2-2%5E2%29=sqrt%289-4%29=sqrt%285%29\"$ .
\n" ); document.write( "Soo the coordinates of the foci are $\"system%28y=-1%2C+x=2+%2B-sqrt%285%29%29\"$ .
\n" ); document.write( "The ends of the latera recta are the points on the ellipse with the same $\"x\"$ $\"%22=%22\"$ $\"2+%2B-sqrt%285%29\"$ $\"%22=%22\"$ $\"approximately\"$ $\"2%2B-+2.236\"$ as the foci.
\n" ); document.write( "For those points $\"x-2=%22+%22+%2B-sqrt%285%29\"$ , so $\"%28x-2%29%5E2=5\"$
\n" ); document.write( "The $\"y\"$ coordinates for those points can be calculated from $\"4%28x-2%29%5E2%2B9%28y%2B1%29%5E2=36\"$ as
\n" ); document.write( " $\"4%2A5%2B9%28y%2B1%29%5E2=36\"$ --> $\"20%2B9%28y%2B1%29%5E2=36\"$ --> $\"9%28y%2B1%29%5E2=16\"$ --> $\"%28y%2B1%29%5E2=16%2F9\"$ --> $\"y%2B1=%22+%22+%2B-sqrt%2816%2F9%29\"$ --> $\"y=-1+%2B-+4%2F3\"$
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