document.write( "Question 334900: Find the equation of the ellipse with x intercepts at (ą6, 0) and y intercepts at (0,ą4) and find the foci \n" ); document.write( "

Algebra.Com's Answer #239980 by jsmallt9(3758) $\"\"$ $\"About$

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The standard form of the equation of an ellipse is:
\n" ); document.write( " $\"%28x-h%29%5E2%2Fa%5E2+%2B+%28y-k%29%5E2%2Fb%5E2+=+1\"$
\n" ); document.write( "for horizontally oriented ellipses and
\n" ); document.write( " $\"%28x-h%29%5E2%2Fb%5E2+%2B+%28y-k%29%5E2%2Fa%5E2+=+1\"$
\n" ); document.write( "for vertically oriented ellipses. In both cases a > b.

\n" ); document.write( "Since the origin, (0, 0) is halfway between the two x intercepts and also halfway between the two y intercepts, then the origin is the center of the ellipse. This makes h = 0 and k = 0.

\n" ); document.write( "And since a is the distance from the center to the vertices on the major axis and since the distance from (0, 0) to (6, 0) and to (-6, 0) is 6, a = 6 and this ellipse is a horizontally oriented ellipse.

\n" ); document.write( "Since b is the distance from the center to the vertices on the minor axis and since the distance from (0, 0) to (0, 4) and to (0, -4) is 4, b = 4.

\n" ); document.write( "In summary we have an ellipse that...

is horizontally oriented (which means that we will use the first of the two standard forms above).
has a center at (0, 0) which means that h = 0 and k = 0.
has a = 6 and b = 4.

\n" ); document.write( "Inserting these values for h, k, a and b into the first of the standard forms we get:
\n" ); document.write( " $\"%28x-0%29%5E2%2F6%5E2+%2B+%28y-0%29%5E2%2F4%5E2+=+1\"$
\n" ); document.write( "This simplifies to:
\n" ); document.write( " $\"x%5E2%2F36+%2B+y%5E2%2F16+=+1\"$

\n" ); document.write( "The only thing left is to find the foci. The foci are found on the major axis they are a distance of c from the center. The center is (0, 0) and the major axis is horizontal. All we need now is the value of c. We get c by using the equation:
\n" ); document.write( " $\"a%5E2+=+b%5E2+%2B+c%5E2\"$
\n" ); document.write( "Substituting our values for a and b into this equation we can solve for c:
\n" ); document.write( " $\"6%5E2+=+4%5E2+%2B+c%5E2\"$
\n" ); document.write( " $\"36+=+16+%2B+c%5E2\"$
\n" ); document.write( " $\"20+=+c%5E2\"$
\n" ); document.write( "Since we are only interested in the positive value for c we get:
\n" ); document.write( " $\"sqrt%2820%29+=+c\"$
\n" ); document.write( "Simplifying the square root we get:
\n" ); document.write( " $\"sqrt%284%2A5%29+=+c\"$
\n" ); document.write( " $\"sqrt%284%29%2Asqrt%285%29+=+c\"$
\n" ); document.write( " $\"2sqrt%285%29+=+c\"$
\n" ); document.write( "Now we have c. We can use this and the center, (0, 0), to find the coordinates of the foci. Since this is a horizontally oriented ellipse, we will add and subtract c from the x coordinate of the center:
\n" ); document.write( "Focus #1: (0+2sqrt(5), 0) or (2sqrt(5), 0)
\n" ); document.write( "Focus #2: (0-2sqrt(5), 0) or (-2sqrt(5), 0)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

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