document.write( "Question 1037746: E is the ellipse with foci at (4,−2) and (4,8) and whose major axis has length 20. \n" ); document.write( "

Algebra.Com's Answer #652553 by KMST(5328) $\"\"$ $\"About$

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So, that is E, but what is the question?
\n" ); document.write( "Write the equation?
\n" ); document.write( "Locate vertices and co-vertices?
\n" ); document.write( "graph?
\n" ); document.write( " \r
\n" ); document.write( "\n" ); document.write( "Both foci are on the vertical line $\"x=4\"$ ,
\n" ); document.write( "so this is an ellipse with foci and vertices on the vertical major axis $\"x=4\"$ .
\n" ); document.write( "The center is the midpoint of the segment connecting the foci.
\n" ); document.write( "That is point (4,3), with coordinates
\n" ); document.write( " $\"x=%284%2B4%29%2F2=4\"$ and $\"y=%28-2%2B8%29%2F2=6%2F2=3\"$
\n" ); document.write( "the averages of the foci coordinates.
\n" ); document.write( "The focal distance, $\"c\"$ , is the distance from one focus to the center:
\n" ); document.write( " $\"c=8-3=5\"$ .
\n" ); document.write( "Since the major axis has length $\"20\"$ ,
\n" ); document.write( "the semi-major axis is $\"a=20%2F2=10\"$ .
\n" ); document.write( "Since the major axis is vertical, the vertices of the ellipse are $\"c=10\"$ units above and below center (4,3), at (4,-7) and (4,13).
\n" ); document.write( "The semi-minor axis, $\"b\"$ , can be calculated using the relation
\n" ); document.write( " $\"a%5E2=b%5E2%2Bc%5E2\"$ .
\n" ); document.write( " $\"10%5E2=3%5E2%2Bc%5E2\"$
\n" ); document.write( "100-9=b^2}}}
\n" ); document.write( " $\"b%5E2=91\"$
\n" ); document.write( " $\"b=sqrt%2821%29\"$
\n" ); document.write( "The equation of an ellipse with major axis parallel to the y-axis,
\n" ); document.write( "center at point (h,k), is
\n" ); document.write( " $\"%28x-h%29%5E2%2Fb%5E2%2B%28y-k%29%5E2%2Fa%5E2=1\"$
\n" ); document.write( "In this case, with $\"system%28h=4%2Ck=3%2Ca=10%2Cb=sqrt%2891%29%29\"$ ,
\n" ); document.write( "the equation is
\n" ); document.write( " $\"%28x-a%29%5E2%2F91%2B%28y-3%29%5E2%2F100=1\"$ .
\n" ); document.write( "The co-vertices are on the horizontal minor axis,
\n" ); document.write( "a distance $\"b=sqrt%2891%29\"$ to the left and rihjt of center (4,3),
\n" ); document.write( "at $\"%22%28%22\"$ $\"4-sqrt%2891%29\"$ $\"%22%2C+3+%29%22\"$ and $\"%22%28%22\"$ $\"4%2B-sqrt%2891%29\"$ $\"%22%2C+3+%29%22\"$ . \n" ); document.write( "

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