document.write( "Question 1170895: Foci at (2, 3) and (2, -1), endpoint of minor axis (4, 1). Find the equation of the ellipse and sketch the graph. \n" ); document.write( "

Algebra.Com's Answer #795833 by greenestamps(13203) $\"\"$ $\"About$

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\n" ); document.write( "The two foci are on the same vertical line, so the major axis is vertical. The standard form of the equation is then

\n" ); document.write( " $\"%28x-h%29%5E2%2Fb%5E2%2B%28y-k%29%5E2%2Fa%5E2+=+1\"$

\n" ); document.write( "The center (h,k) is at (2,1) -- halfway between the two foci; a is the semi-major axis and b is the semi-minor axis.

\n" ); document.write( "The semi-minor b is the distance from the center (2,1) to the given point (4,1), so b=2.

\n" ); document.write( "The distance from the center to each focus is c, where a, b, and c are related by

\n" ); document.write( " $\"c%5E2+=+a%5E2-b%5E2\"$

\n" ); document.write( "The distance from the center to each focus is 2, so c=2. Then

\n" ); document.write( " $\"c%5E2+=+a%5E2-b%5E2\"$
\n" ); document.write( " $\"4+=+a%5E2-4\"$
\n" ); document.write( " $\"a%5E2+=+8\"$

\n" ); document.write( "Now we have the center (h,k), and we know a^2 and b^2, so we have all we need to write the equation:

\n" ); document.write( " $\"%28x-2%29%5E2%2F4%2B%28y-1%29%5E2%2F8+=+1\"$

\n" ); document.write( " \n" ); document.write( "

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